Is my ex-boyfriend responsible for paying any part of bills that we acquired together?

When parties are married, it is somewhat easier to show that debts such as that are "joint" and that each of you are responsible. Without being married, you would likely have to show he undertook some sort of legal responsibility to share in the bill. There is no clear answer without knowing more. If you believe he promised to share the costs, you may want to consider filing suit in magistrate court and see what a judge says. There is no guarantee of success though.