It can be the switch. A wire is unlikely as the same two wires are used going to the motor for UP as well as DOWN. The polarity is just reversed and the switch does that.
The other possibility is that the motor is "stalled" because of a problem inside.
I wouldn't spend any time on the wiring. After you remove the inner door panel, you can peel back the liner and about in the middle area of the door you'll find a connector like this:
That is the motor feed. If you pull that, and feed 12 volts from a cigarette lighter plug or whatever to it, you can drive the motor either way, based on which pin you make +12, and which pin ground.
You can also take it apart and put a meter on it, or better, push the meter probes in to the back of the connectors while it's connected. Then operate your window switch and see what goes to the motor. If you get no voltage in the "up" position, but you do get down voltage, then it's the switch.
If you get voltage in both directions, it's likely the mechanism itself.