Hi Scott!
We've experienced situations where we moved a site to a new server, a new IP address was assigned and the rankings dropped for a month or so, then leveled out and we regained all original rankings back. However, like Ryan stated, we did not make any changes to the url structures, page names, etc. in this case. Also using a site like he referenced helps a lot. You can also use this site to find out what other sites are on the server you are hosted on: http://www.ipfingerprints.com/reverseip.php
We have also migrated another site to a new server, keeping the same IP as the previous server and the rankings did not fluctuate at all.
I would think that the answer lies in "HOW" you perform the migration and be sure to "measure twice, cut once". As the move could be very smooth if done properly, however it could turn into an absolute disaster if rushed and done improperly. If you haven't already, check out this website migration guide. Study it. Know it. Implement it. Hope this helps!