Note, the servo library (Servo - Arduino Reference) mentions some restrictions when you have more than 12 servos.
Just to be sure, I imagine you already power the servos separately with a common ground, and aren't trying to power the servos from the Arduino 5v pin.
As a trouble shooting method, I would hook each servo up one at a time until I found where it broke, rather than going full speed and try all 18 at once.