Yes, it's possible to get pregnant during your period. You'd have to have a very short menstrual cycle, which is the time from the first day of one period to the first day of the next period, or a tendency to have long periods.
An egg can survive in your fallopian tube for about 24 hours after it's released from the ovary. And if sperm are present in your fallopian tube at that time, one of them may fertilize the egg and lead to a pregnancy. If the egg isn't fertilized, it's shed along with your uterine lining during your period.
A typical menstrual cycle is about 28 days long, but some women have cycles as short as 22 days. If you have a shorter cycle, you could ovulate just a few days after your period ends. And since sperm can survive in your reproductive tract for up to three to five days, if you meet your partner during your period, it's possible that the sperm may still be hanging around when you ovulate, ready to fertilize the egg.
It's also possible to mistake breakthrough bleeding — spotting between your periods — for a regular period. So you may be bleeding right at the time of ovulation, when you're most fertile. If you were to conceive at this time, you might believe that you were having your period when you got pregnant.
If you had unprotected sex during your period and are concerned that you may be pregnant, look out for symptoms like mild lower abdominal cramping, spotting (implantation spotting, a fairly common cause of bleeding once the egg has been fertilized), breast tenderness and moodiness. These symptoms can occur as early as two weeks after ovulation.