Is this a modded forum? AFAIK phpBB closes all connections and even cleans up vars rather primly after itself, vanilla - but mods and addons may not be as well made.
I would say moderately modded. It has extreme style, ezmod, simple subforum and a few other small mods but not a radical PHPBB layout mod. I had this problem before as I stated and they increased the MySQL client support. I am thinking that BOT's are not helping matters either as I have seen one IP tie up 30 or more connections. Site is still down as hoster is working issue. I am looking for a more permenant fix here and host advised of PHP code not releasing clients properly.