I can only assume it uses up all the material in it after so many miles and exhaust. If one went bad I can only assume the other one is on it's way out.
Catalysts aren't consumed by the reactions they facilitate. In theory, a catalyst should never wear out. In practice, however, they eventually become contaminated and their effectiveness goes down. This contamination can be accelerated by poor mixture control (either too rich or too lean) or by other contaminants getting into the combustion chamber (e.g. burning oil or coolant).
Your car, as you said, has 3 catalytic converters. The usual arrangement would be one for each bank of cylinders and then a third further downstream after the headers have merged into one exhaust pipe. If one bank of cylinders were a little off, then the catalytic converter for that bank would be contaminated the fastest, and the downstream cat would have minor additional contamination, but the cat for the "good" cylinder bank wouldn't have any problems. It all depends on where the contaminants are coming from.
At 240,000 miles, it's probably a good time for a thorough tune-up anyway though. Things like having the injectors rebuilt, making sure the MAF sensor is clean, that the fuel filter is changed, and that the fuel pressure regulator isn't getting weak, etc. Might be overkill for a daily beater, but I tend to err in that direction with my own cars.
ZV