Create a new project in one of the older versions of AE, make a new composition, and put a solid layer in it. No effects or anything.
Does that open?
If you let the CC app remove your old versions when you installed CC 2015.3, remember: your CC subscription includes CS6, CC, CC 2014, CC 2015, and CC 2015.3, so you can install a couple of older versions for testing purposes (and for issues like this). I'd recommend installing CC 2014 and, if you have room, CC 2015 in addition to CC 2015.3. (Here's how to install older versions.)