Can a bulged capacitor prevent a CPU from booting?
I found that one of the capacitors has bulged slightly. I want to know
  if this bulged capacitor is causing my CPU to not boot?


It sure can, even if the cap still retains its capacitance, the ESR is most likely too high. This will cause more ripple than normal and could interfere with digital signaling (the power supply needs to be clean). Pull the cap off the board. Check the cap with an ESR meter (should be under 1Ω or even lower) and replace. I would replace if it was bulging, make sure it has a higher voltage level than listed on the cap, and the same or more capacitance. This may or may not fix the problem. 

Check the power with a Digital Multi Meter and check the AC ripple in mV it should be lower than 25mV and hopefully in the 10-5mV range.