My medical school offered us the chance to get an MPH concurrent with our medical degree (DO in my case), and so long as we were in good standing in the medschool, the tuition for the MPH was waived. It didn't make us graduate late or anything.
But just to be sure, before I started it, I ran it by the HPSP coordinator for my area. I was told that it was ok to do, but if I failed a medschool class and had to repeat it, it would be assumed to be because of the additional coursework I had chosen to take on, and the penalties would be... severe. They didn't elaborate, nor did I ask them to do so.
So I just got the DO/MPH without failing any classes and all was well.