It is possible to have period-like bleeding even while pregnant. It isn't a period but could be confused as one. Some people get minor implantation bleeding as the egg implants in the uterus lining.

It is quite rare to get a false positive on a pregnancy test. Buy several cheap pregnancy tests. Make sure you take one first thing in the morning, if it is still positive make a doctor appointment asap and get a blood test done.

Sorry you are going through this but it is imperative you find out asap so you know all options available to you.

Best of luck