Confused about my pports.?

Hopefully someone knows more about this. My parents are from Belfast Northern Ireland. I was born in the U.S. I already have two pports. I have a U.S. pport and an Irish pport. I'm planning on going to graduate school at Queens University in Belfast. However, my cousin in Belfast says that it may be cheaper or something if I get my British pport. I forget why but she said I should get it anyways. I guess because it's in Northern Ireland. Is that possible to have three pports? Shouldn't I have a British one in the first place since my parents were born in Belfast? Wouldn't that put my U.S. citizenship in jeopardy if I get a British passport? Basically, I need to know if I can infact get a British passport if I already have an Irish and American passport.