Good question. However, I don't think it's possible as long as the messages stay within the Facebook system. However, if they email you direct from their account, its quite possible you'll be able to get their IP....unless they're using Gmail.
Well, I can't say what message tracing service FaceBook will provide. For sure you'll have to justify the search. If the other person has used some cunning (for example, created a fictitious new facebook account from a public library) then this might not lead anywhere.
Maybe you could take this up with FaceBook and let us know what happens. I'm sure there are forum readers who would like to learn from your experience.