I'd suspect the battery getting bad. More specific, the internal (series) resistance has increased?

This would explain the behavior so that when you do have high current consumption (WiFi radio downloading movie from internet and CPU fully loaded decoding the video), the series resistance drops the battery voltage the N8x0 sees and thinks the battery is dying... In other scenario using with small load the series resistance doesn't cause much drop with the battery voltage, thus it seem working normally. Series resistance does not affect the battery capacity, so the standby time should appear normal.

Once the tablet says the battery is dying and possibly shuts down, if you reboot it does it still say the battery is empty or does it come back showing good charge level? If so, then this should prove the theory above and you should try new battery.

By the way, in my N810 I have not seen this problem when downloading and watching movie. (Maybe you can try this using the WorldTV99... which should cause similar load scenario than watching YouTube flash movies... That would either confirm or rule out the possibility that the flash player is causing this.)