Definitely, I had a pretty severe car crash that limited my physical activity for over six months when I was at USAFA. When I got healthy(ish) I was at ground zero for fitness (we're talking 10 sit-ups, 1 or 2 real push-ups). I did sit-ups and push-ups every day and by the time I took the USAFA PFT about two months later I was able to do about 40 push-ups (perfect ones, I had 5 more that weren't quite good enough to count) and I did the USAFA PFT max for sit-ups (95 in 2 minutes). Hope that gives you some encouragement.
The key for me was to do as many as I could of both at least 3 times a day and for push-ups, I did as many as I could and then as many MORE as I could on my knees
Good luck! It also helps to have a partner. I had roommates that would do them with me to help keep me motivated