You will no doubt laugh at The Mighty Konrad for asking such a basic question, but here it is:
 
My cursor no longer snaps on even time divisions (like 1/8, 1/16, etc.).  The grid on the Control Bar is lit up (blue) and set to 1/16, but when I place notes they end up at weird times like 3:3:28 or whatever instead of snapping to the right place.  After drawing notes, I have to go back and quantize.
 
Similarly, I can't select even beats or measures and have to resort to Select by Time.  When I paste, I have to type in the time: I can't just place the cursor on a bar line because the paste will be off a few ticks.
 
Surely there is something little thing that got unchecked somewhere...?

Maybe the strength of the snap got changed?

Did you happen to enable snap by instead of snap to?

I'd also check your PRV snap settings. I've been confused by the same thing more than once until I noticed my PRV snap settings were on and different to the main snap.

Thanks, guys.  This is an issue in Staff and Track views.  I'll look around some more and report back.