Yes, it is possible, if you were off it between prescriptions long enough to release an egg. For the same reason, doctors advise women that when starting the pill it is possible to get pregnant during the first month's packet. If your gap between the two prescriptions was only a few days for your period, you are probably ok.
If the new package of pills that is started is the same prescription as the first, the birth control will be effective provided that you are taking it as prescribed. If you are changing medications or have a gap in dosage, you will want to use a backup method for seven days.
Birth control pills are not completely 100% guaranteed no matter how long you have been taking them. If you were taking them and now you are pregnant it is obvious that the answer is yes you can get pregnant while on birth control pills. It would appear that you are an example of the fact.