Advice from a math grad student.

Before you invest substantial time in a research project, make triply sure nobody has solved the problem already. Often there's an easy but time-consuming way to check this: if there's a paper that such a result is guaranteed to cite (such as the paper in which the problem was first posed), then read the abstracts of all 100+ papers citing it on Google Scholar or something. If there are a few such guaranteed citations, do this for all of them. If something looks vaguely related (most of the papers won't be), then skim through the whole paper.

I've been bitten by this 1.5 times. The first time around was my very first research project in graduate school; I assumed that the professor posing it had already performed this check. Once we had actually proven a result, we discovered that the same result was proven 14 years ago. Moreover, a few years later, the same author had solved the extension we were going to pursue next.

The second time around, I read a Wikipedia article which stated that no improvements to an upper bound were found. This was for semi-legitimate reasons because it required putting together two results in a trivial way to see that an upper bound already exists. By the time I realized this, we were already on the writing-a-paper stage; fortunately, our upper bound was better than the existing one, so we were saved.