Here are my thoughts on the error... No poles correct me if i'm wrong please ;)
This error doesnt matter which page your on, what browser your using etc. The MySQL database which essentially runs the site (forums, videos, pictures, etc) probably has some sort of limit of the number of connections it can accept at one time.
So, when there are a lot of users trying to connect... (or maybe even certain users trying to make many simultaneous connections See my 'Speed up Firefox' thread) this is causing the error.)
Hope this helps.