Yep! I have the same problem, also with Hostgator
I'd say its down to too many guest connections from bots. Normally, my forum ambles along with perhaps 5 connections at the most. Then all of a sudden the connections will spike causing the site to crash.
As Hostgator said, it seems to only way for us to fix this problem is to upgrade our packages to a Dedicated Server
After talking to one of their sales team, you can get a dedicated server for just under $20/month if you don't want to use their cPanel. ($45 with the cPanel).
Once I leave university and get myself a job, this is probably the route I'll be taking.