I actually did this during college - went from DIII to DI. It was somewhat of a disaster, and I ended up going back to my DIII school. I wish I never left, though I might not have known that had I not tried it.
I have also coached in DIII and DI. I am currently a DI coach, and despite that, I am far more fond of DIII and the way athletes are developed. I think DIII athletes, on average, have a better experience because more of them have an opportunity to be successful, boost their confidence, etc. When I coached in a really high level DI program, I felt really bad for the kids at the back of the team. The (very high profile) head coach treated those kids like they were total scrubs, was kind of nice to them to their faces, but behind their backs, he would say horrible stuff about them. They didn't know that, but I'm sure they felt it in the way he interacted (or didn't interact) with them. I often thought about how much better off those kids would be in the DIII programs in which I had previously coached.
I think that a lot of DIII stars think they could be so much better if they were running DI, and that may be the case sometimes, but I also think there's some element of DIII being responsible for or aiding their success, whether or not they realize it.
It completely depends on the schools and your individual situation, but I think DIII undeservedly gets a bad rap. As a coach, I'd MUCH rather coach at a high level DIII school than a medium to low level DI school, despite the fact that DI is generally considered to be more glamorous.