I'm having a weird and similar problem...I have a 9v power supply that I'm trying to use with a brand new Due board.
The board works fine when I use the USB power supply.
But when I plug in the 9 volt barrel connector the Due loses power.
When I test the voltage at the VIN pin it shows 9 volts (so clearly the 9 volt power is getting through ok). The 5 volt pin measures at 1.23 volts and the 3.3 volt pin measures at 0.45 volts.
With reference to BJHenry's post above I tried two different sized barrel connectors on different power supplies with no difference to the results.
As an aside, I tried to find the 9-12 to 5 volt regulator on the board. From the schematic (https://www.arduino.cc/en/uploads/Main/arduino-Due-schematic.pdf) I was looking for IC2 - an LM2734Y but I couldn't find it on this layout (http://i.imgur.com/7YtQ3xh.png). I would have expected it to be a fairly large device like the 5 to 3.3 volts regulator IC4 which is a NCP1117ST33T3G. However, I think I must have a later revision board because my Due (an original) has a different layout and seems to have a large device next to IC4 doing the 12 volt regulation. I am currently wondering if it's working properly???