The Absolute Maximum Operating Voltage of the ATMega2560 is 6.0V. The same applies to the ATMega8u2 that does the USB to Serial on the Arduino Mega board.
Recommended maximum voltage is 5.5V
So yes, you can run it at 6 Volts, but you are right on the edge, so I wouldn't recommend it.
Possible risks are a decreased lifespan of the Arduino, and there might be some stability issues.
If the application isn't safety critical, and you don't mind having to replace the board after it failed in, say, two year of operation, it could be an acceptable risk (if there are no easily available alternatives)
PS you might want to measure you power supply. It may say it's 6V but could give out 6.2V. The output voltage of some supplies also vary with the load that's across it.