User:octal has a suggestion:
It looks like some of the "too many files open" problems are being caused by the large numbers of session files being used by PHP. Is the site maintaining a session for every anonymous visitor? This might cause the problem. (See www.php.net/manual/en/ref.session.php for info on custom session handlers, if you need to store so many sessions but the filesystem is not robust enough, you can move session handling into a database).
Either that, or something in the site is dependant on having a lot of concurrent files open. Good luck with wikipedia. -- octal