Andy_Langton - 9:01 pm on Feb 24, 2013 (gmt 0)
Google uses the DNS just like everybody else, so other than the short period where there would be to separate DNS records for the same site (i.e. at the time you change the records to point to your new host) there is no duplication.
It's possible, of course, to manually request a domain name via an IP that does not appear in the DNS, but that would be pretty rare and is certainly not an activity that search engines undertake.
Of course, even if they did so, it would still be the same content under the same domain name so there would still be no possibility of issues.
The only possible problem is if you serve the site on the old host under something other than your name (e.g. via an IP) but then, if you have a proper canonicalisation procedure in place, this would redirect to the canonical host anyway.
I think this is one you shouldn't need to worry about. That said, checking for duplication is a good idea generally, and if you found an old host cropping up for some reason, add it to your canonicalisation procedures.