This problem looks like it has been resolved (finally!) I am just posting this here in case it saves someone else from the hell I just endured.
I was notified a week or so ago that my site would be moved to a new machine. I have a VPS with several domains on it. Friday morning I got an email confirming the sites had been moved successfully. One of my domains would come online, for 4-20 minutes and then go offline for 4-20 minutes. When the site was offline I could not even FTP in. However, the IP addresses were accepting pings and traceroutes without a problem, nothing weird in the server logs, no crashes, no reboots, nothing. All the other domains were fine.
The site was doing this for nearly 4 days! I opened several emergency tickets with call backs, and chatted with numerous techs. Each time, they would run some configuration, or tell me to up my memory allotment, see that the site was up (for the moment) and then close the ticket and act like it was all better. Then 20 minutes later the site would go down again. Then they even moved my account again to another machine, but the problem persisted.
Finally today we got a call that when they moved the site the first time, our unique IP (needed for our SSL) was still partially associated with the old machine. So the IP would point to the old machine with nothing on it, but still had the ability to accept pings. Then the new server would grab the IP address for a few minutes and serve files like nothing was wrong. Then the old machine would grab it, site goes down, site goes up, site goes back down - on and on like that for FOUR DAYS.
It appears to be resolved now. I have 3.5 hours of uninterrupted up-time since this last change. So far so good. I just wanted to share my experience.