Some of the capacitors are dual voltage. One side of the cap feeds the fan motor and the other side feeds the compressor. Look at cap and see if the case looks distorted, as this is a sure sign of a bad cap. Is the compressor running as you mentioned a humming and lights flickering? Look at the start relay as I have seen spiders and ants get in the relay blocking one side of the 220v causing a single phase condition to the compressor, hence humming when trying to start. If this happens the overload will take out the compressor until it cools, then compressor will try and start again. If the cap is ok, and relay is ok, and the compressor windings read continuity with an ohmmeter, but the compressor will not start I would put a hard start kit on the compressor. This is a secondary capacitor which gives the compressor an additional boost to start. I would try cutting off the thermostat, and killing the breaker to the unit for about a minute. Then turn on the breaker, then turn on the thermostat, set to cool and see what it does. I have seen control boards get a glitch and resetting power would correct the problem, especially after a storm when you get a surge from lightning. If this doesn't fix the issue check as I discussed above. Worst case is bad compressor, or out of gas and low gas safety has it taken out.