Why would the neutral wire in my bathroom's outlet box have voltage?
Before I explain my answer, turn off your circuit breaker! If the wires are truly switched, there is a hazardous shock/fire hazard present and you need to turn off the circuit breaker right now! If the options below don't resolve your situation, call an electrician for help. The problem you're describing can be very dangerous.

There are two possible scenarios here.

1) The receptacle circuit hasn't changed - If you haven't changed anything (wiring, additional receptacles, new receptacles), then it sounds like there is a short to ground. Your circuit breaker should be tripping though, which is odd. You will have to check the wiring to and from all receptacles and the load center.

2) The receptacle circuit has changed - I believe this is the most likely scenario. Have you replaced wiring or receptacle in that circuit? Keep in mind that something you changed that you thought was on another circuit may be connected to this one. Check all of your connections from the load center to the receptacle. 

To answer your theoretical light bulb question, you are correct. A standard incandescent light bulb (not LED) will turn on if placed between hot/neutral, neutral/hot, and even hot/ground, ground/hot. It will not work with ground/neutral or neutral/ground because they have the same voltage potential and therefore can't carry current (V=iR). Please note that although it would work between hot and ground, this is a code violation and is very dangerous because you will be passing current through the grounded enclosures and you will also be creating a shock hazard. 

You can test receptacle wiring with something like this: Receptacle Tester. However, this will only work if the circuit has power, which as you pointed out, your GFCI keeps tripping.