Too Many Connections Error—Shouldn't It Reset?

A friend of mine has been getting a "Too many connections" error on his vBulletin forum (the entire site is throwing up a database error; nothing works) since he tried using a MySQL backup script. I occasionally got this error on my own board from increased traffic, but it always went away and things returned to normal. This is not the case for him, and his host won't fix it on the weekends.

What, exactly, has to be done here? Does the database somehow have to be reset? Why are things *still* (even with the backup script disabled) going wrong?