Can I ever know if my question has been answered before? I am working on a project that is very interesting to me, but I am unsure about whether the research question has been asked before. What is the best way to find out if my research question has already been answered before?
I am going though the dissertation process myself, and have discovered that your advisor is always the best way to go. In theory, your advisor is an expert in the field, and ought to be current with the literature in the field he is guiding you through. Your committee also has this job. Trust them - if you publish on something that's already out there, their reputations as cutting-edge researchers would be on the line. 

Still, your advisor and committee are human, and part of the job to make sure your topic hasn't been hijacked already is yours. In my case, being primarily a teacher, research isn't my forte, and my advisor is close to retirement; this meant that we discovered very late that the problem I had been attacking for my dissertation is in literature under a topic name unknown to either of us nor to my committee (compressed sensing). Even after having found my topic addressed, major open questions still remained, and some approaches I had may still be considered original. In short, the point here would be to keep looking in every corner even after you've picked a topic and started writing. The most sure way to "look in every corner" would be to find the most inspiring research papers you're using for a project and check to see who has referenced them, and check to see who has referenced the papers these works themselves referenced.