If there is an Apple store near you try taking it in there and connecting it to one of their computers with your cable. If that works then it's probably your computer.
Are you running it off a PowerMac G4? I had all sorts of problems connecting my iPod to it (and, judging by the forums, a lot of people did). The solution that worked for me was, when I had a problem I would power down the PM, unplug the cord, pop the case open, press the PMU reset button, wait 10 seconds or so, then plug it in and go.
There seem to be some problems with the firewire ports on some PM G4s that this will help fix (at least until it happens again