I think technically you're probably nearly always still drunk the next morning after a heavy night - in the sense that you'd fail a breath test.
But it feels very different from being drunk the night before, because the environment is totally different. A bit like being in a pub and only realising just how pissed you really are when you go to the "quiet room" (i.e. the toilet....)
I don't really ever get drunk nowadays, I've become a health nut, but my worst ever was staying in an almost-empty hotel in Margate for seven weeks during the winter (when Margate is completely dead). We got drinking with the hotel staff, I had a whole bottle of Jack Daniel's (or thereabouts) and woke up face-down in a pool of vomit the next morning. Went to the toilet, splatter-painted all three walls of the cubicle with spew (sorry about this...), felt utterly dreadful for the whole day and couldn't stand even the smell of JD for at least 10 years.
Oh, and then there was the time I vomited in the bath. When I was in it. It's all coming back to me now.
And that's about as rock'n'roll as I've ever got.
(Sorry, I guess those are hangover stories more than drunk-the-next day stories...)