It can be done, if you are really super-super frugal, and if your stipend is reasonably generous. I have never had student loans but I have managed to save about $37k, not counting any family contributions, and while paying all my bills. If I can save $37,000 in grad school with no debt, I'm sure if I had had $20k in debt, it would have been destroyed by now.
But the catch is, you REALLY have to make sacrifices (no car, shared housing, don't eat out/travel much), and of course your stipend has to be decent. If you're stuck making $16k/year, it won't happen.