Fun fact: apparently they will reject you for the force if you score too highly on the exam. Don't want the enforcers too smart, I guess.)
This happened once, and the excuse they gave is that smart cops would get bored and quit and they would have wasted money training him/her. I don't know if that holds water, but the courts think so. It's certainly not common practice for local departments to not hire a person that is too smart.
As for the OP, I would trust a random cop (local, state or fed) slightly more than I'd trust a random non-cop.
This nugget has some truth to it. The phys test is actually a IQ test (though the mental part is just a 10 min interview). The rumor/theories is that a high IQ cop will over think things to his or others detriment. Not to say that they want dumb cops as it is even worse (see incidents like Charleston).
If I have seen and heard of a few people failing the phys test after scoring too high. Of course they don't flat out tell you that but instead come up with something like you failed the polygraph or something in your background etc.