Is it possible to be pregnant even though I got my period?

No. If you had a menstrual period, you are not pregnant. On the other hand, it is possible to be pregnant and have vaginal bleeding that may seem like a period. For example, loss of an early pregnancy will cause bleeding. More worrisome is the possibility of bleeding from an ectopic pregnancy. If it's possible you are pregnant, see your doctor right away.
No. When you become pregnant your ovaries are suppressed and will stop cycling. Therefore you won't have a period. Some women have bleeding while pregnant for various reasons and mistake this bleeding for a period.
No. We have all heard of women who "had their period" all the while they were pregnant. This is biologoically impossible. What this usually means is that there was some other reason or complication of the pregnancy that caused irregular bleeding. So often someone may not know they are pregnant because they "had their period" when in actuality they are having pregnancy related bleeding.