Capacitor Banks for Current Spikes
Just some ideas for things to check first:

If you're using something like an ATX power supply, I'd suggest you look for its specs, or if not available the ATX power supply specification, and check the section about capacitive loads.  (I believe too high a value can cause instability in the supply itself.)

I would also recommend (if you're not doing so already) using completely independent wiring for the 0V and Power wires for each servo - that is, say three sets of thick wire, each going from one servo to the supply. Instead of having one thick pair of wires which feeds all of the servos.  (Just in case there are any problems with resistance in sections of the wiring under heavy current.)

Finally you might want to fit small decoupling capacitors to suppress noise spikes, fitted close to the servos themselves. I'm talking about any reasonably low ESR (equivalent serial resistance) ceramic capacitor or similar, these are commonly used for motor noise suppression.  (Surprisingly electrolytic capacitors aren't so good for this, they often have higher resistance.)

Actually... I wonder if https://electronics.stackexchange.com/ would be a good place to search for answers too as there are bound to be people there used to motors and power supplies...