Sadly (or should I be happy that Google is this awesome? Not to mention the rate of scientific progress!), all I really had to do to come up with an answer was perform a Google search for "learning transcranial magnetic stimulation". The first hit, a ScienceDaily page (Ruhr-University Bochum, 2011) page, lists some journal references (Mix, Benali, Eysel, & Funke, 2010; Benali et al., 2011) that effectively say (in laymen's terms), "Yes!" Here's ScienceDaily's summary (I'll let you read the rest at the source):
What sounds like science fiction is actually possible: thanks to magnetic stimulation, the activity of certain brain nerve cells can be deliberately influenced. What happens in the brain in this context has been unclear up to now. Medical experts have now shown that various stimulus patterns changed the activity of distinct neuronal cell types. In addition, certain stimulus patterns led to rats learning more easily. [Emphasis added.]
To show how hard I'm working to really earn that bounty of yours, here's the summary from the second Google hit, a review article on this very topic (Reis et al., 2008; again, plenty more info at the source):
In summary, the scarce studies performed so far point to the encouraging conclusion that noninvasive brain stimulation can contribute to the understanding of mechanisms underlying motor learning and motor memory formation and raise the exciting hypothesis that this increased understanding could in the future result in the development of new strategies to enhance specific stages of learning and memory processing in healthy humans and in patients with brain lesions (see chapter by Gerloff et al.). [Gerloff et al. doesn't appear in the references; emphasis added.]
Edit: I see from your comment that you want me to explain the mechanism. Since you offer a bounty, I'll play along and dance for it a bit. Here's another excerpt from Reis and colleagues (2008, page 6 of 16):
It would be theoretically possible to facilitate motor learning processes in which [the primary motor cortex] is involved by enhancing excitability in the “learning” [primary motor cortex (Pascual–Leone, Valls–Solé, Wassermann, & Hallett, 1994)] or by decreasing excitability in the “resting” [primary motor cortex (Schambra, Sawaki, & Cohen, 2003; Plewnia, Lotze, & Gerloff, 2003)], but see also Wassermann[, Wedegaertner, Ziemann, George, & Chen (1998)]. The intrinsic intracortical mechanisms by which these oversimplified models may operate remain to be identified ([Perez & Cohen, 2008; Daskalakis, Paradiso, Christensen, Fitzgerald, Gunraj, & Chen, 2004; Koch, Franca, Mochizuki, Marconi, Caltagirone, & Rothwell, 2007;] see for discussion chapters by Walsh et al., Di Lazzaro et al., Berardelli et al.). [References unavailable for these three discussion chapters; emphasis added.]
You may want to read the review further for yourself or ask follow-up questions if this is an unsatisfactory explanation of the mechanism.
Since that review was a little hesitant to conclude conclusively, and it's five years old now, I'll also throw in a brand-new bit of original research (Wall et al., 2013) that demonstrates the supportive trend continues:
Conclusion: These preliminary findings suggest rTMS does not adversely impact neurocognitive functioning in adolescents and may provide subtle enhancement of verbal memory as measured by the CAVLT. Further controlled investigations with larger sample sizes and rigorous trial designs are warranted to confirm and extend these findings. [Emphasis added.]
$\uparrow$ That $\uparrow$ was the fifth Google hit! Also, the seventh (Jelić et al., 2013) says it's not a placebo effect!
Going a little wide, transcranial direct current stimulation (which is somewhat different from magnetic stimulation, though maybe not in ways that concern you) works too (Fields, 2011; Kincses, Antal, Nitsche, Bártfai, & Paulus, 2004; Fregni et al., 2005; Nitsche et al., 2003; Ohn et al., 2008; Flöel, Rösser, Michka, Knecht, & Breitenstein, 2008; Chi & Snyder, 2012)—you can even try it yourself for $249 or £179! I just had to follow a few links from the 14th Google hit (Mims, 2012) to find all that. I'm a little shocked (pardon the pun) at how legit this actually seems...
Edit (by Fizz): Note that a more recent (2017) meta-analysis of tDCS studies found significant problems with their p-curves, making the authors of this meta-analysis conclude that
Using a p-curve analysis, we found no evidence that the tDCS studies had evidential
value (33% power or greater), with the estimate of statistical power of these studies being
approximately 14% for the cognitive studies, and 5% (what would be expected from
randomly generated data) for the working memory studies.
Also (16th Google hit), it looks like the U.S. National Institutes of Health Clinical Center is currently recruiting participants for their study, "The Effect of Transcranial Magnetic Stimulation on Learning With Reward in Healthy Humans", so maybe you could even get Uncle Sam to pay you (for testing magnetic, not electrical stimulation)! However, if I'm reading that right, it's a study of learning disruption, which is another potential (and maybe not-so-desirable) application of TMS, according to the 18th hit (De Weerd et al., 2012).
Edit: Reis and colleagues (2008) also review this disruptive process somewhat; I get the impression that disruption is much easier than enhancement to achieve with magnetic stimulation (not sure whether this applies to direct electrical current as well). It seems TMS has been used mostly to knock out brain function temporarily in localized areas to simulate the effects of brain lesions. Use of TMS to enhance excitability rather than inhibit it seems to be the more recent innovation, and maybe a separable effect (i.e., it may be possible to excite the brain using TMS without causing inhibition).
In anticipation of the next question, "Is it safe?" I feel I should add that this is definitely not my area of expertise – I got a B- in biopsych as an undergrad – so again, I'll let the experts speak for themselves (Poreisz, Boros, Antal, & Paulus, 2007), but as a fan of dystopian sci-fi, I will at least say that the short-term dangers seem like they could've been much worse! Same for the ethical ramifications, more or less (Hamilton, Messing, & Chatterjee, 2011). It's reassuring to see so much thought going into this already; I might otherwise be worried about how far behind the times we seem to be here! BTW, these last revelations come thanks in part to the 33rd Google hit (Oremus, 2013). I'm afraid no amount of tDCS will allow us to keep up with Google...
P.S. From what I've seen in skimming some of these, it doesn't have anything in particular to do with the limbic system (which includes the amygdala). I saw a brief explanation suggesting it affects myelination, which is basically the process by which the brain speeds up electrical signal transmission. Myelin is a sort of segmented coating for the axons of neurons, which can be quite long individually. Myelin allows electrical signals (action potentials) to jump across myelinated segments quickly, rather than propagating relatively slowly across every millimeter in the normal fashion. I also saw some indication that transcranial electromagnetic (didn't note which) stimulation might speed up other aspects of neural construction, such as the rate at which connections form (not just the rate at which they upgrade from the neural equivalents of a dial-up modem to fiber-optic).
Benali, A., Trippe, J., Weiler, E., Mix, A., Petrasch–Parwez, E., Girzalsky, W., ... & Funke, K. (2011). Theta-burst transcranial magnetic stimulation alters cortical inhibition. The Journal of Neuroscience, 31(4), 1193–1203.
Boggio, P. S., Ferrucci, R., Rigonatti, S. P., Covre, P., Nitsche, M., Pascual–Leone, A., & Fregni, F. (2006). Effects of transcranial direct current stimulation on working memory in patients with Parkinson's disease. Journal of the Neurological Sciences, 249(1), 31–38. Retrieved from http://www.tmslab.org/publications/154.pdf.
Chi, R. P., & Snyder, A. W. (2012). Brain stimulation enables the solution of an inherently difficult problem. Neuroscience Letters, 515(2), 121–124. Retrieved from http://ingienous.com/wp-content/uploads/2012/05/snyder-9-dots-paper.pdf.
Daskalakis, Z. J., Paradiso, G. O., Christensen, B. K., Fitzgerald, Gunraj, & Chen. (2004). Exploring the connectivity between the cerebellum and motor cortex in humans. Journal of Physiology, 557, 689–700. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1665103/.
De Weerd, P., Reithler, J., van de Ven, V., Been, M., Jacobs, C., & Sack, A. T. (2012). Posttraining transcranial magnetic stimulation of striate cortex disrupts consolidation early in visual skill learning. The Journal of Neuroscience, 32(6), 1981–1988. Retrieved from http://www.jneurosci.org/content/32/6/1981.long.
Fields, R. D. (2011, November 25). Amping up brain function: Transcranial stimulation shows promise in speeding up learning. Scientific American: Mind & Brain. Retrieved from http://www.scientificamerican.com/article/amping-up-brain-function/.
Flöel, A., Rösser, N., Michka, O., Knecht, S., & Breitenstein, C. (2008). Noninvasive brain stimulation improves language learning. Journal of Cognitive Neuroscience, 20(8), 1415–1422. Retrieved from http://www.researchgate.net/publication/5548486_Noninvasive_brain_stimulation_improves_language_learning/file/5046351db1c615613a.pdf.
Fregni, F., Boggio, P. S., Nitsche, M., Bermpohl, F., Antal, A., Feredoes, E., ... & Pascual–Leone, A. (2005). Anodal transcranial direct current stimulation of prefrontal cortex enhances working memory. Experimental Brain Research, 166(1), 23–30. Retrieved from http://www.researchgate.net/publication/7745540_Anodal_transcranial_direct_current_stimulation_of_prefrontal_cortex_enhances_working_memory/file/9fcfd50a3483033595.pdf.
Hamilton, R., Messing, S., & Chatterjee, A. (2011). Rethinking the thinking cap Ethics of neural enhancement using noninvasive brain stimulation. Neurology, 76(2), 187–193. Retrieved from http://wernicke.ccn.upenn.edu/~chatterjee/anjan_pdfs/Neurology_2011_HamiltonMessing_Chatterjee.pdf.
Jelić, M. B., Stevanović, V. B., Milanović, S. D., Ljubisavljević, M. R., & Filipović, S. R. (2013). Transcranial magnetic stimulation has no placebo effect on motor learning. Clinical Neurophysiology, 124(8). 1646–1651.
Kincses, T. Z., Antal, A., Nitsche, M. A., Bártfai, O., & Paulus, W. (2004). Facilitation of probabilistic classification learning by transcranial direct current stimulation of the prefrontal cortex in the human. Neuropsychologia, 42(1), 113–117. Retrieved from http://www.researchgate.net/publication/9011312_Facilitation_of_probabilistic_classification_learning_by_transcranial_direct_current_stimulation_of_the_prefrontal_cortex_in_the_human/file/d912f50597d44df446.pdf.
Koch, G., Franca, M., Mochizuki, H., Marconi, Caltagirone, & Rothwell. (2007). Interactions between pairs of transcranial magnetic stimuli over the human left dorsal premotor cortex differ from those seen in primary motor cortex. Journal of Physiology, 578, 551–562. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2075160/.
Mims, C. (2012, March 8). DIY kit overclocks your brain with direct current. Technology Review: View. Retrieved from http://www.technologyreview.com/view/427177/diy-kit-overclocks-your-brain-with-direct-current/.
Mix, A., Benali, A., Eysel, U. T., & Funke, K. (2010). Continuous and intermittent transcranial magnetic theta burst stimulation modify tactile learning performance and cortical protein expression in the rat differently. European Journal of Neuroscience, 32(9), 1575–1586.
Nitsche, M. A., Schauenburg, A., Lang, N., Liebetanz, D., Exner, C., Paulus, W., & Tergau, F. (2003). Facilitation of implicit motor learning by weak transcranial direct current stimulation of the primary motor cortex in the human. Journal of Cognitive Neuroscience, 15(4), 619–626. Retrieved from http://www.researchgate.net/publication/10710628_Facilitation_of_implicit_motor_learning_by_weak_transcranial_direct_current_stimulation_of_the_primary_motor_cortex_in_the_human/file/79e4150ae2024da0f0.pdf.
Ohn, S. H., Park, C. I., Yoo, W. K., Ko, M. H., Choi, K. P., Kim, G. M., ... & Kim, Y. H. (2008). Time-dependent effect of transcranial direct current stimulation on the enhancement of working memory. Neuroreport, 19(1), 43–47. Retrieved from http://diyhpl.us/~bryan/papers2/neuro/Time-dependent%20effect%20of%20transcranial%20direct%20current%20stimulation%20on%20the%20enhancement%20of%20working%20memory.pdf.
Oremus, W. (2013, April 1). Spark of genius. Slate: Technology: Superman. Retrieved from http://www.slate.com/articles/technology/superman/2013/04/tdcs_and_rtms_is_brain_stimulation_safe_and_effective.html.
Pascual–Leone, A., Valls–Solé, J., Wassermann, E. M., & Hallett, M. (1994). Responses to rapid-rate transcranial magnetic stimulation of the human motor cortex. Brain. 117(4), 847–858.
Perez, M. A., & Cohen, L. G. (in press as cited in Reis et al., 2008). Mechanisms underlying functional changes in the primary motor cortex ipsilateral to an active hand. Journal of Neuroscience.
Plewnia, C., Lotze, M., & Gerloff, C. (2003). Disinhibition of the contralateral motor cortex by low-frequency rTMS. Neuroreport, 14, 609–612.
Poreisz, C., Boros, K., Antal, A., & Paulus, W. (2007). Safety aspects of transcranial direct current stimulation concerning healthy subjects and patients. Brain Research Bulletin, 72(4), 208–214. Retrieved from http://www.researchgate.net/publication/6377248_Safety_aspects_of_transcranial_direct_current_stimulation_concerning_healthy_subjects_and_patients/file/9fcfd50a34830ccb05.pdf.
Reis, J., Robertson, E. M., Krakauer, J. W., Rothwell, J., Marshall, L., Gerloff, C., ... & Cohen, L. G. (2008). Consensus: Can transcranial direct current stimulation and transcranial magnetic stimulation enhance motor learning and memory formation?. Brain Stimulation, 1(4), 363–369. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2621080/.
Ruhr-University Bochum. (2011, January 29). Learn more quickly by transcranial magnetic brain stimulation, study in rats suggests. ScienceDaily. Retrieved February 10, 2014 from http://www.sciencedaily.com/releases/2011/01/110128121629.htm.
Schambra, H. M., Sawaki, L., & Cohen, L. G. (2003). Modulation of excitability of human motor cortex (M1) by 1 Hz transcranial magnetic stimulation of the contralateral M1. Clinical Neurophysiology, 114, 130–133.
Wall, C. A., Croarkin, P. E., McClintock, S. M., Murphy, L. L., Bandel, L. A., Sim, L. A., & Sampson, S. M. (2013). Neurocognitive effects of repetitive transcranial magnetic stimulation in adolescents with major depressive disorder. Frontiers in Psychiatry, 4(165). Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3859914/.
Wassermann, E. M., Wedegaertner, F. R., Ziemann, U., et al. (1998). Crossed reduction of human motor cortex excitability by 1-Hz transcranial magnetic stimulation. Neuroscience Letters, 250, 141–144.