I have an Arduino UNO that I plugged into a 12V 3A supply. It connects to an Adafruit motorshield that runs two steppers and a relay board with 4 relays. With the jack plugged I seem to get fluctuating power after about five minutes of use. However when I plug the USB in everything works fine. At first I thought it was a heat issue as the board itself was fairly toasty, but that doesn't make sense as it still will run on the USB.
The regulator is overheating and shutting itself down to prevent damage to the board. When using 12v input, you can't pull very much current from the 5v rail without it overheating (it;s a linear regulator - so that 7v times however much current you're drawing is dissipated as heat, and that little regulator (which doesn;t have the tab tied to a large copper area of the PCB to act as a heatsink like best practices suggest, even though components on the board could have been located soas to do that)