Most of these problems are likely caused by a bad battery but some, like yours apparently, have a bad charge/power regulator which will cause the same symptoms. I'm betting it'll kill that battery too by either repeated overcharging or undercharging maybe even the wrong charge voltages.
I've seen other signs that point to weak power circuitry design (or more likely just one or two crap Chinese capacitors) in the Nexus 10
Looks to me like a Job Opportunity replacing batteries in out of warranty Nexus 10's .... I'm not a "User" I'm a Senior Level Certified Electronics Technician .... Hopefully Samsung did a little better design than the iPad which takes an afternoon and a heat gun just to change a simple battery (Which wouldn't be so bad if Apple actually passed along the savings but they don't so they can maintain their ridiculous 30% Margins)
Personally I'd rather have a slightly thicker device that has an easily replaceable battery, less heat related problems, and no backlight bleed inherent in frameless/bezel-less LCD's which are all the costs of a super thin design ....