I've experienced some strange problems with modems. Some similar to the one you described. Something even weirder happened to me: the computer would spontaneously turn itself on. For some time I had to unplug the AC cable to stop it from returning from the dead.
Until I found out it was the modem. Just had to unplug the ethernet cable to stop the problem.
Anyway, try to unplug the modem for some time, while you're working or whatever. See if the problem persists. It's a long shot, but...