As Chuck pointed out in the comments, it's important not to take a metaphor too literally. Comparing our memory to a mailbox may have some validity, it is not true that our memory can "fill up"-- i.e., that we have a limited capacity for knowledge in general. No matter how old or how many facts you have learned, you will always be capable of learning new ones.
In the NYT article, prominent aging research Timothy Salthouse points out this research is controversial:
"It seems unlikely that any single study, regardless of its quality, will be able to resolve the issue. We don't know how to measure storage capacity."
Additionally, your question asks whether we risk "slowing down our brain functions" if we study too much. This study is about age-related deficits, and there is no evidence that simply learning more information will reduce our memory capacity.
The study also is referring only to working memory, and not long-term memory. While some researchers believe working memory is simply a subset of attended items in long-term memory (e.g. Cowan, 1999; Anderson J.R. et al., 2004), others believe they are unique constructs (e.g. Baddeley, 2000). It is important not to conflate the two.
While learning new facts cannot reduce our overall memory capacity, it may affect our ability to retrieve specific items. This is known as memory inhibition. For instance, recalling semantically related words can inhibit retrieval of a target word (Anderson, M.C. et al., 1994).
In sum, learning new information may prevent us from retrieving older information through inhibition, but it is not related to overall memory capacity, or to the age-related deficits mentioned in the NYT article. We don't risk "slowing down our brain functions" by studying too much.
Anderson, M. C., Bjork, R. A., & Bjork, E. L. (1994). Remembering can cause forgetting: retrieval dynamics in long-term memory. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20(5), 1063. PDF
Anderson, J. R., Bothell, D., Byrne, M. D., Douglass, S., Lebiere, C., & Qin, Y. (2004). An integrated theory of the mind. Psychological review, 111(4), 1036. PDF
Cowan, N. (1999). An embedded-processes model of working memory.Models of working memory: Mechanisms of active maintenance and executive control, 62-101. PDF