Doubtful. I have a 1st generation iPod at work that has a totally dead battery. Since I use it as an emergency hard drive I don't care. As soon as I disconnect it it stops functioning...but when I reconnect it it reboots and all is well. I suspect something else is going on...or you got a bum drive.
Hmmm....if it functions normally whilst on an AC charger, then chances are you have a dead battery. However, if it continues this behavious whist being charged via AC, then you most likely have a dead HDD. Seek out an AC charger, then give it a whirl. The difference between USB/Firewire charging and AC charging is that the AC gives a much higher amperage [double at worst!], so the iPod can function at full capacity whilst being charged.