The short answer is no. It is not possible to have a true menstrual period during pregnancy. Your hormone levels during pregnancy will change to prevent you from menstruating, and it is not possible for your body to shed its entire uterine lining while maintaining a pregnancy.
It is, however, possible to have menstrual-like bleeding for a variety of reasons during pregnancy. Women who report having periods during an otherwise normal pregnancy are usually experiencing a phenomenon that is sometimes called decidual bleeding, in which a small part of the uterine lining might shed for the first few months of early pregnancy at the time that the woman would otherwise have had her period. Decidual bleeding is not a true menstrual period, but it can look similar enough to cause women experiencing it to not realize that they are pregnant until fairly far along in the pregnancy.
Another possible explanation for having bleeding that looks like a period in very early pregnancy is implantation bleeding, which is spotting that may occur around the time of the first "missed" menstrual period. Implantation bleeding would occur only during the first month of pregnancy, however.
Note, however, that bleeding during pregnancy should always be reported to a doctor in order to rule out miscarriage or other complications. Decidual bleeding does occur in some women but is fairly rare. Implantation bleeding usually lasts only a day or two. So seeing a doctor is your best bet for ruling out miscarriage and figuring out the reason for your bleeding during pregnancy.
Note that "decidual bleeding" is not a common medical term; your doctor may simply refer to this bleeding as "first-trimester bleeding."
Pairman, Sally, Jan Pincombe, Carol Thorogood, and Sally Tracy. Midwifery: preparation for practice. Australia: Elsevier, 2006. Page 625.