Yes it is possible.
But not with PoE. The board can be powered using the GPIO. Here is a pin diagram showing the pins for power at the top.
Maximum permitted current draw from the 3v3 pin is 50mA.
Maximum permitted current draw from the 5v pin is the USB input current (usually 1A) minus any current draw from the rest of the board.
Model A: 1000mA - 500mA -> max power draw: 500mA
Model B: 1000mA - 700mA -> max power draw: 300mA
You can read more on elinux.
