Yes, there is a small possibility of becoming pregnant during your periods. Most women often mistake a slight bleeding that takes place with ovulation as their period, and having unprotected sex during this time can significantly increase your possibilities of becoming pregnant. You should consult with your doctor and you can get more information here http://www.epigee.org/menstruation/sex.html.
What I heard is a bit different than the above answers. It is a NO. If one is on periods, the uterus draining out all the preparation (blood, ovum etc). Next chance of pregnancy will be after a weak (may be more depending on one's own cycle) at least. & I read that sperm will not live that longer. Any how these days everybody advices safe sex.
Ok... DSRao is wrong. Completely unequivocally wrong. You CAN get pregnant while on your period. I'm living proof of that. My mother was under the same assumption when she naively thought the same thing as DSRao... She thought it was a fluke, and got pregnant 2 times because of having unsafe and unprotected sex during her period. Luckily she was married to my father during both times or who knows what could have happened.
Typically a woman ovulates around 14 days before the start of her period. So if you have an average 28 day cycle, you would most likely ovulate on cycle day 14. A woman?s cycle, though, is not always predictable. Even women who normally have regular cycles can have an off cycle and ovulate earlier or later than normal. If you ovulate very early in your cycle, for example, on cycle day 10, it would be possible to conceive if you had intercourse while on your period.