It's very possible! I got pregnant while on Depo, and no I didn't miss any of my appointments for the shot. Never late for them. And I still ended up pregnant. It's very possible to get pregnant on depo, or any birth control for that matter, which is why a back up contraceptive is always helpful. In fact the percentage of effectiveness for birth control is 99.9% however it's only 99.9% effective with the use of a back up contraceptive such as a condom. 1 in 30 people become pregnant on birth control because it is not used properly.
I would suggest you take a home pregnancy test, The first response is a highly effective and accurate test to use.
The weight gain alone doesn't tell much because weight gain is highly common on depo. (and can happen even if it hasn't before) as for movement, they can also be in your head because it is possible to convince yourself. If you still assume pregnancy, go to a clinic or take a test. If you're feeling movement you should have no problems showing positive on a pregnancy test.
It is possible since you had unprotected sex while off of the shot... It might be that you weren't far enough along for a urine test to pick up the hCg levels. I would say take a home test and then let your doctor know how the test turns out. If they refuse to do anything more for you then maybe you should consider changing doctors. Good luck!