My ipod (5 80g video) refused to charge for a while. It would boot into charge mode (on cable connection) and quickly restart, each time getting less and less far into the process, shorter time between resets. I concluded from this that the battery was near dead and the cable wasn't providing power or the charging circuit wasn't recognising it.
Here, it seems like the iPod kept turning on because voltage was detected from the cable, but the battery was very low, not charging and actually being discharged due to the power used by attempts to boot.
I neglected to fix this and months later tried to charge it (with a different cable, I still haven't tested the (possibly faulty) other one) and all was well, but the run time is quite deficient.
The 5th generation iPods can charge via FireWire and USB. Does this new cable charge with the same method? If yes, then the old cable is probably faulty. If no, it's possible that the iPod cannot charge via one of the charging methods.
Which of these is most likely?:
battery deteriorated from being stored with no charge?
This is most likely. You were discharging the iPod well below its hardware-based shutoff point. Then the battery self-discharged further during storage. It's quite possible the battery became overdischarged and damaged.
battery shorted somewhere so it drains quicker than usual? I never had the false connection recognition that is discussed here...
This is easy to test. Does the runtime get a lot shorter if the iPod is left turned off for a day or so?
charging logic prematurely deciding the battery is charged, and rockbox misrepresenting the charge?
You can observe charging via the battery debug screen. The most important thing to watch for is voltage reaching about 4.2V. After this, current should start decreasing until it reaches a low value and charging ends.