I've noticed a high-pitched buzzing on my bass, on the E and A strings. It seems to happen the most at certain frets, and I'm sure it's not just regular fret-buzz caused by bad fretting technique. What could be causing it?
Not as far as I can tell.

A couple things to add:

It still happens when palm-muting.

The strings look a lot closer to the other frets (for example, if I'm fretting the fourth, the string looks very close to the fifth fret); I'm starting to think the action got lowered somehow, or there's something wrong with the neck....
Your strings might have been knocked and bent slightly. Other than that or a neck/fret issue I can't think of any other potential causes.

After watching closely, I don't think the strings are vibrating against the frets. Is there any way it could be a problem with the nut or bridge (and would these still occur while fretting or palm-muting?)? Also, what do you mean that the strings would be bent? Like, the core is bent? If so, would there be any tell-tale signs?
Well, you said high pitched and palm muting, could it be a accidental harmonic?
I think you might be adding in some artificial harmonics on accident.
It happens when I normally pick too; I don't think it's a harmonic, but I'll look into the possibility.