Given that I don't want to do real work, I figured, hey, I'll take this out of the wall!
Long ago, when I owned a different house, my dad once told me that the little slot in the backs of these medicine cabinets was for the disposal of razor blades. You'd use the blade until it was too dull, but that left it too sharp for safe disposal, so you would just push it through the slot and it would fall down into the wall. I was never sure whether he was pulling my leg or not. But ...
I guess he wasn't. Look at the font on those. I'll bet they could be dated in archaeological layers. I honestly am tempted to do just that. It would have to tell me when this apartment was retrofitted into the carriage house.
Marlin High Speed razor blades were a product of the Marlin Firearms Company of North Hampton, Connecticut. They started making razor blades in 1936 and phased them out sometime in the 50's or 60's (I'm not sure from my brief research). Those were obviously the last ones used and disposed of in the wall -- but upon inspection of the bottom of the pile, it appears that somebody really liked Marlin blades.
There is one Gillette Blue Blade, with corner indents dating it to no earlier than 1933 (oh, Mr. Google, is there nothing you can't tell me?) It has arrows printed on it indicating the direction it was to be inserted into the razor; these were introduced in 1948.
Also at the bottom: one Silver Star Duridium blade; these were advertised around 1949 to 1950, attracting the attention of Linus Pauling in a letter dated June 14 of 1949. Also a Blue Star "FINEST BLUE STEEL" blade; those are still available today, making dating somewhat more difficult, but appear to be around 1950's vintage.
I am totally guessing 1949 as the year this apartment was first occupied. It is left as an exercise to the reader to calculate from the dimensions of the pile, the dimensions of a typical razor blade, and the number of days a blade could safely be used, just how many years this mostly-Marlin-using occupant lived here.