This morning we woke up to the bathroom lights not working. Investigation reveals that the culprit is the GFCI that serves two bathrooms and the laundry room.
It still pops in 10-15 seconds with no draw whatsoever on anything that appears to be connected to the circuit. (I haven't gone through the house looking for stray outlets on the line, though.)
My first thought was a bad GFCI but a couple of other things come to mind also:
1) Two days ago we replaced every bulb on the circuit with LED lights.
2) It's pouring rain, something that happens a few times a year here. We certainly aren't aware of any water leaks but I could imagine water coming down a vent pipe and then going somewhere.
Do I just replace the GFCI and hope it works? Is there some means by which I can test if there is an electric problem? (With the GFCI popped wouldn't it show infinite resistance between the hot and the neutral and between the hot and the ground?)
Edit: Replaced the GFCI, I get exactly the same behavior. I've loosened every bulb on the circuit, no change. Notably I'm getting very low (the highest I observed was 2v) but non-zero voltages downstream of the GFCI when the wires are hanging loose. That's with a decent multimeter so it's possible this is just induced current from somewhere.
Edit again: It apparently is water related somehow. In the days after the rain it was slower and slower to trip, now it's behaving correctly. I can't see how water could get anywhere near the wires, though, let alone to any junctions. If it happens again I can try disconnecting things to find the culprit.