I have two potentiometers plugged in, and I am viewing them in the debug window. One of them is red and oscillating slightly due to noise (which is fine), but the other is black and stuck at 4095. Does this mean the pot is dead?
4095 is right at the top of what a poti should be reading, and usually won’t be reached by a working one. Try different ports and check the external wire and plug for any shorts, but if it still doesn’t work it probably is dead and you’ll just have to get a new one.
Ok, it turned out that on one extreme of the pot, it was reading 4095 for an entire angle range of 45º. It looks like it’s misbehaving on the extreme. Thanks 
My guess would be that it’s been turned past the intended physical stop at some point. But if your use can stay well out of that zone you won’t need a new one!
Potentiometers can only handle ~250 degrees of motion. They have crescent-shaped attachment holes for that reason, to allow you to adjust the potentiometer’s mounting angle so that it can fully measure the device’s motion. Like Kevin said above, something may have broken the physical stop at some point. Try not to damage the stops.