I had my 6510 in my luggage for a couple of weeks when I moved and when I went to revive it it was dead-dead-dead. The battery was dead of course but charging it doesn't seem to help. I've left it plugged in for over 24 hrs but nothing seems to change: the red led cycles off and on maybe every 3-5 seconds and the display shows and HP logo very dim. when I disconnect the charger there's just no sign of life whatever.

I wondered if there was something like a non-rechargeable backup battery that might need to be replaced. Any other thoughts?

ok, cleared. I think there is some sort of backup battery and, if it runs down, the unit will no longer charge. I was able to bring mine back by popping in a charged battery and putting my dead battery in a friend's to charge. After 5 minutes my backup battery had a bit of charge and i was able to charge my own battery again.

Odd design!