The W5100 draws 200-300mA of current. That chip alone means the onboard regulator is burning (11V-5V)*200mA = 1.2W.
It is very likely the chip, the ATmega328, and whatever else your project has connected is causing the regulator to go into thermal shutdown. If you have a power supply that outputs 9V or a better external regulator to provide 5V you may get more stable operation.