I believe the issue has been resolved. They are still looking into the exact cause, but there was a problem with another account on the shared hosting server that BWD hosted accounts are on. Here is the reply from tech support:
I think someone's site was hacked, and they keep doing something to overload the server. I'm trying to narrow which site is causing the problem...
Not completely reassuring, but other responses were more assuring, that our sites should not go down again at this time.
Thanks for your patience and understanding.
Let me know if you are still experiencing difficulties.