MichaelMeissner:
It is a problem if the chip isn't designed for it. However, if you have an official nano, then it in theory should be supported. Quoting from: http://arduino.cc/en/Main/ArduinoBoardNano.The Arduino Nano can be powered via the Mini-B USB connection, 6-20V unregulated external power supply (pin 30), or 5V regulated external power supply (pin 27). The power source is automatically selected to the highest voltage source. The FTDI FT232RL chip on the Nano is only powered if the board is being powered over USB. As a result, when running on external (non-USB) power, the 3.3V output (which is supplied by the FTDI chip) is not available and the RX and TX LEDs will flicker if digital pins 0 or 1 are high.
There is a discrepancy between what the product description page says and what the actual schematic shows for the nano 3.0. Until proved otherwise I will assume that the schematic is correct, unless someone has other proof to share?
If you look closely at the nano 3.0 schematic ( http://arduino.cc/en/uploads/Main/ArduinoNano30Schematic.pdf ) you will see that the FTDI chip is NOT powered directly from the USB +5vdc (which is named VUSB), but rather powered from the same +5vdc source that the 328p chip is, named +5V. Which means that the FTDI chip is powered up no matter if it's plugged into a PC or not. So simply if the 328p is powered up so will the FTDI chip be. The older arduino boards using a FTDI chip operated the same, the FTDI chip is powered up even if not plugged into a PC USB.