Old Tank Syndrome could be considered the opposite of New Tank Syndrome, but the effects are the same, shock to the fish.
In new tanks, fish are introduced and must go through the phases of cycling, and dealing with high levels of toxins. These toxins such as ammonia, nitrite, and nitrate cause stress and could be fatal to most fish.
In older tanks, the lack of scheduled maintenance on the tanks could cause build-ups of fish waste that is undetectable in test kits. Depletion of certain minerals and the change in chemistry caused by the nitrification process could alter the GH, KH, PH of the system. If left unchecked, and unmanaged, these levels can stray vastly from your tap water parameters. Usually Old Tank Syndrome doesn't rear it's ugly head, until a water change is performed. At that time, the change in GH, KH, PH, and whatever else that has built-up, or depleted takes a huge shock, and the fish is stressed, and can lead to death.
These are the theories at least, no first hand experience in this, as I don't allow my tanks to get to these conditions. Prevention is the best cure.