My mom is a girl/boy identical twin, and my boyfriend is a twin, but not identical. I am 6 weeks pregnant. Twins run on both sides. Is there a possibility that I could have twins?

Not identical. Identical twins mean they are exactly the same dna a girl and boy can't be identical they have to be the same sex the possibility of you having twins is much higher than the general population as twins run in both your families having non-identical twins means that the mother releases two ova at the same menstrual cycle identical twins come from one fertilized ovum hoping you have what you wish fo.
Girl/ Boy twins. Are not identical, they at fraternal, coming from 2 seperate fertilized eggs. There is a chance you have twins but having a mother who was a twin and a boyfriend who was a twin makes no significant difference in your rate of having twins. Raising one child is difficult, twins is at least twice as hard. Chances are you are pregnant with a singleton, ultrasound will tell you.