For example, in a game earlier, I was playing Eve and ulted as I died. The ability appeared to not go on cooldown, so when I came back up, I tried to use it (failed). After about the normal time, I could use it again, but I had no idea how long the cooldown was - after the next use it showed the cd as normal.

I've seen this happen on other champs without dying - for example this happened to me yesterday with Irelia's Q. It appeared to be usable again after I used it (thought I last hit a creep, must have missed it). Could not use it, didn't show cd, and ~15 seconds later or whatever it was usable again and showed the cd normally.