I think it completely depends on who you are. Everyone is unique, obviously. I know for me, I got pregnant on the pill, and I still continued to get my period. I did not find out until I was three months pregnant and miscarrying that I was pregnant at all (it was my first year of college and I seriously thought the extra weight was just the "freshman 15" :p )
my aunt, however, also got pregnant on the pill, and she missed her period that first month. She was so upset because she didn't know if the fact that she had taken birth control would have hurt the baby in any way, but alex was born nine months later, perfectly healthy, and is now 7 years old and still in perfect health.
I would say that even if you're on birth control, and you've had unprotected sex, and you think you might be pregnant for whatever reason, even if your still getting your period... It never hurts to find out for sure. I know it's gotta be rare, but hell, it happened to me. Maybe i'd still have my baby today if i'd have known and not continued to take the pill all the way through :(