Very strange. I know I've been online days that we've celebrated having 100k visits (or close) with no problem.
It also depends on what parts of the site are being used, as certain scripts demand more database abuse than others, I suppose.
I've experienced the "too many connections" error a few times since I've been here, but never one that lasted over a few hours like today's.
Too bad Kosta wasn't online today, he could have run the processlist in MySQL and probably figured out exactly where the issue was coming from.