The server was unreachable for 15 minutes yesterday (not down, but not possible to connect to it), and then this morning I got a "server down page" at 3:45am.
The server is reachable again, obviously since I am posting this. Apparently the datacenter where the server is located had a major network problem, which took an hour to resolve. They house thousands of servers there. I am not sure if all were effected, but other customers were posting about being down on some of the hosting forums. The company has a very good reputation, and this is only been the second prolonged full network outage since we have been with them.
This is unrelated to the slow page load problem we were experiencing for the first week after moving to the new server.
I am waiting for a detailed explanation of what happened, but now that everything seems back to normal, I might go back to bed for a few hours and then check for their update on what happened to their network..