Depression is characterized by emotion dysregulation (e.g., Joormann & Vanderlind, 2014). This means that depressed patients have difficulties decreasing (downregulating) their negative affect and increasing (upregulating) their positive affect. This may be a consequence of cognitive distortions and deficits (Joormann & Vanderlind, 2014) as well as maladaptive emotion goals (e.g., the goal to increase/maintain negative affect; Millgram, Joormann, Huppert, & Tamir, 2015).
Many kinds of therapies seek to improve (directly or indirectly) the ways depressed patients regulate their emotions, including cognitive-behavioral therapy (Beck, 2011) and cognitive bias modification (Hallion & Ruscio, 2011), which includes memory therapeutics (e.g., aiding recall of positive autobiographical memories or reducing the impact of negative memories; Dalgleish & Werner-Seidler, 2014) and attention training (e.g., Papageorgiou & Wells, 2000).
These therapies are effective/efficacious at reducing the cognitive deficits associated with depression and thus facilitate more adaptive emotion regulation (i.e., less rumination, more positive reappraisal, more self-distancing). This leads to improved mood (decreased negative affect, increased positive affect; e.g., Aldao, Nolen-Hoeksema, & Schweizer, 2010).
So, yes, depressed patients can experience happiness via establishment of more adaptive emotion regulation skills. Moreover, positive affect is associated with activation of many different brain regions (Lindquist et al., 2015), which makes it unlikely that depressed patients have a total inability to experience happiness.
Interestingly, positive memory recall may not be effective for depressed patients as a short-term mood repair strategy. In fact, it may make them feel even worse (Joormann, Siemer, & Gotlib, 2007). Depressed patients may not find positive memories to be intrinsically rewarding, perhaps due to processing disfluency and/or dysfunctional reward circuitry (Chen, Takahashi, & Yang, 2015).