My take on all this:
My definition of “Skeptic” is someone who is cautious and always logical about accepting unsupported data as being fact. Not just a naysayer.
By my definition, no, there's no such thing as being too skeptical. There's no such thing as being too dedicated to making fact and logic based decisions. The only minor addendum I would point out, is that I consider it to be an act of irrationality to ignore that there are situations where we must act before we can gain the desired amount of factual information needed to make ideal decisions.
However, I have seen that the label of skeptic, has become generally seen to be a positive trait, as opposed to the same person accepting the label of “stubborn,” or “prejudiced,” or “close-minded.” Hence, lots of prejudiced, self-blinded, irrationally certain people, CALL themselves “skeptics,” and provide lots of examples of the kind of “skeptic” that the opening poster seems to be concerned about.
So no, there’s no such thing as being too much of an ACTUAL skeptic, but there is certainly such a thing as allowing people who don’t qualify as skeptics, to hide their prejudices and other thinking defects behind that label.