For the first 2 years it should be possible, but you'll need someone else to take care of the dog during the second two years. It's pretty difficult to get all your rotations in Bethesda during the MSIII year and even if you do you'll have required 24 hr shifts (at least for surgery and peds you'll have several during the rotation). You'll need someone who can feed him/her for those overnight shifts, and possibly have to board the dog during away rotations. I know in San Antonio you can have smaller dogs in the housing we're put up in, but at other locations they don't allow dogs.
Also, I would ask an MSI about the schedule. It's different from what I had - on my schedule during MSI/II, it would have been possible especially if you're not a lecture goer. But the schedule has changed a bit, and I don't know exactly how many breaks they get. You can find the current MSI schedule here:
http://131.158.7.207/oea/ms1.html
(Deans time is student free time)