I can only tell you what my experience was after I recieved a similar email from Host Gator at the end of May.
After the migration I started noticing a higher bounce rate on my sites. Watching the sites carefully I saw that the page load times were extremely high, but only on an intermittent basis. The times for the same page would vary between a little over 2 seconds to well over 20 seconds, sometimes within a 5 minute window. I never saw the sub 2 second load times I'd seen on the older server.
When I contacted Hostgator they aknowledged that it was a problem on their end and that they were working to fix it. However, they weren't able to fix it. Towards the end, the last week or so, they back tracked and said it was either my site, or that it was just the nature of a shared hosting environment. It was not the nature of the previous shared hosting environment, nothing had changed with my site, and from what I was reading online at the time, I wasn't the only one having issues after the migration.
I finally gave up at the end of July, almost 8 weeks later, and moved to another hosting company.