Turns out squirrels had chewed off the protective cover of one of the wires, and it was touching another
They also tossed in another modem/wireless gateway for free, just to get us to shut up and not bother them again.
When three people tell you everything is plugged in and no they're not running any bit torrents and YES the damn LIGHT is STILL TURNING RED EVERY FIVE MINUTES for a whole week straight, you can assume that something really is messed up and they're not incompetent nincompoops after all.