I would not use 5V for the ENC28J60. The absolut maximum rating according to the data sheet is 3.6V
Some other interesting informations from the ENC28J60 data sheet:
Supply Voltage: Min. 3.1V, Typ. 3.3V, Max. 3.6V
Operating Current: Typ. 120mA to 160mA, Max. 180mA
And here some facts from the STM32F4DISCOVERY:
The STM32F4 microprocessor is powered by only 3.0V (that means all outputs are also only 3.0V)
There is a voltage regulator (LD3985M33R) on the board which indeed outputs 3.3V, but there is a diode (BAT60J) after it with a voltage drop of 0.3V => 3.0V.
And the max. current the voltage regulator can deliver is 150mA. According to the data sheet of the DISCOVERY board it can provide max. 100mA for external sources.
Conclusion:
To operate the ENC28J60 you need an additional 3.3V source. Maybe another voltage regulator (5V -> 3.3V).