I finally found the solution to a problem that was killing me. Before I found it, I asked an SO question about it, and had even started a bounty on the question. Can I now post a self-answer and, if no better answers get posted before the bounty expires, award myself the bounty?
No, you can't assign the bounty to yourself.
However, you can add your own answer and accept it so that other people will know that's what solved the problem.
If the answer is good enough, it will get upvotes then it may even earn a separate bounty at some point from another user who finds it exceptionally useful.
Worth to mention that in the past it was possible to grant ourselves a bounty (example), resulting in +[rep] to show besides the answer but no actual reputation gain which was just confusing. It was removed back in July 7, 2011. (Four months before this question was asked)