It is strange that all three units have the same problem. I am assuming that when you tested your three units that you did so by attaching them to your camera. If all three are bad, it could be that the problem is in your camera. Let's disregard the ready light for the moment because the ready light could be bad or your flash is only partially charging. As an experiment, turn on one of your SB-600's and after allowing time for it to charge, short out the contact pins on the flash shoe. If it flashes, it could be the camera at fault.
The SB-600 is an older unit. If you have three of them, I will assume that you are using them professionally and they have had heavy consistent use. As a result, your speedlight's capacitors may have reached the end of their life span.
Let's look at what may be happening. By charging, it means that the batteries in your speedlight are supplying a low voltage to a capacitor bank that charge to a higher voltage that is sufficient to fire the flash tube. All capacitors deteriorate over time and use and will lose their ability to charge and maintain a charge. Given the age of your equipment, that may be the case. If you have the knowledge and equipment, I would replace the capacitors on one of your units and see if it works. If not, I would send one of the units into Nikon for a free repair estimate. They can be reached at 1-800-645-6687 9AM-8PM EST, Monday to Friday.
on Feb 14, 2015