I've been using AE for about 15 years now, and have never seen this behavior before. Just within the last few weeks, when I try to drag media into the project, about 40-60% of the time, it doesn't work. No errors or anything, just nothing happens.
It's not that I'm trying to bring in incompatible media, because if I try again, sometimes it will then work. Or if I bring the same media in using the import dialog, it always works.
I can't find any common element to the problem, as it can happen with any type of media - audio, photos, video. Any ideas?
I'm on a recent model Mac Pro, with version 10.10.3 of Yosemite, and the latest version of AE CC2014. I tried deleting the prefs already.
It sounds like you face a little detective work to determine what on your machine changed recently. Anything that got updated. They would be good starting points.
Could be one of the gigazillion graphics overlay bugs that plague Yosemite where elements are not detected and the respective system functions not triggered. I'm afraid that's way beyond AE, at least until the next update may provide built-in fixes or workarounds.