This has happened to three (out of 50) Dues. They work fine when connected via USB to a computer. But they do not work when 12v is applied to the Due 2.1mm power connector, or when 12v is applied to the Vin pin.
It is not an issue with the power supply because I've tried several. I also see this is a common problem with other Arduino models.
BTW, for me this is a problem out of the box, it's not a situation where the Arduino was working, then later not.
Could this be a voltage regulator problem, or something reparable?