Reading this article made me wonder if a satellite can be turned off and then back on. I've never really thought about how satellites are maintained and serviced. You can't exactly send IT up there to fix things.
At it's most basic, a satellite will have two systems. A highly robust command and control system with a fairly omnidirectional antenna. And then the more complex system that handles the payload(s). So yea, if the payload system crashes, you can restart it via C&C.
nasa seems to reboot things so I don't see why not. When they do though I think its really nail biting while they hope to hear from it again when it boots up.