Is it possible to be pregnant and have a period

No, not technically... however you can be pregnant and have some bleeding, which some women mistake as their period. Often, but not always, the "period" will be a bit different from the norm... less flow, days may be off a bit, etc.. Most women will have no periods, some will have some spotting or bleeding in the beginning, and a few will have some bleeding throughout or for most of their pregnancy.

If you think pregnancy is possible, either try a test or wait until the next period would be due and see what happens. Test at that time if need be.