How does one protect an IP address?

Okay, the following questions are going to sound pretty uneducated because I don't have an in-depth knowledge of computers. My roommate and I have had struggles with our internet tonight, as it has been constantly cutting out (this happens every week at least once). Tonight, for the second time ever, we received a message saying someone else is using the same IP address. As an aside, we have been struggling with staying under the allotted amount of GB's per month which has always confused us because we do not torrent, nor are we home very often. Is it possible to copy someone's IP address to steal their internet, and if so how the hell do we stop it? Could this be the reason why we go over our limit every month? Sorry if this makes no sense.