After more trouble-shooting, I found that the window would work or not work depending on how I moved the wires where they went from the vehicle frame into the door. Sure enough, there was a completely broken wire in there. Here are a few pictures from that part of the project - it's really a very simple fix if you have one of the electrical connecters and some electrical tape on hand.
The wires pass through a rubber piece as they go from the frame to the door. Just peel back one end of this to get at them. If you will be working on any broken wires, make sure to disconnect the vehicle battery first
The broken wire
I trimmed approx. 3/8ths of an inch of insulation off the end of the break and connected the two with a crimp connector
The connector, crimped with vice grips
Once I had connected the two ends of broken wire, I reconnected the vehicle battery and tried the window. It worked . . . but only one way! Now the window would go down easily, but not up at all. Finally, after more wrassling with the wires, I discovered a second broken wire - apparently there are different wires for moving the window up or down.
The second break
I used a second crimp connector to fix the other break, and then wrapped everything in electrical tape to hold it together inside the rubber casing.
So, there's a simple fix for a power window that won't work. I recommend checking these wires first (and the wires going from the vehicle frame to the driver's door) if you notice that the window works intermittently, rather than cutting out all at once.
You can tell if power is getting to the door if you have lights inside the window switch. Our excursion has two lightbulbs inside each window switch, and I could see there was a connection problem because they would blink on and off when I moved the wires around.
Hopefully this is helpful to someone - it's a simple fix for what could be very costly to have professionally repaired, and besides, it's a lot of fun to work on this stuff!