I've seen at least three things described as "Old Tank Syndrome". All of them relate to poor water change habits and poor tank maintenance/technique.
First, marsh mentioned, is the accumulation of nitrogenous waste. Without water changes, it would not be possible to remove nitrates (excluding plants), so they will build up to toxic levels eventually. Sometimes when this happens the fish will adapt to the polluted waters, but weaker fish and new fish wont be able to adapt and die. Whether or not a large water change will affect this problem isn't clear.
The second problem often rolled into 'Old Tank Syndrome' is the tendency for tanks to acidify over time. The nitrification process that breaks down ammonia into nitrate also acidifies the water. Over time, this can potentially drop the pH very low, potentially into hazardous levels for the BB and livestock. The effect is pronounced in poorly buffered and overstocked tanks. A large water change in this case could potentially be hazardous to the fish.
The third case occurs when tanks are 'topped off' with tap water after a significant amount of water has evaporated. Remember than when water evaporates, only pure water is removed; the minerals in the water remain. Adding water adds both water AND more minerals, so that the resulting tap water has more minerals than initially. Short term, this isn't that much of a problem, but if this is repeated several times, the mineral content of the water will increase steadily. This can unfortunately go on for some time, leading to increasing salinity. This by itself isn't much of a problem, as most fish can adapt to it. The problem comes when you do decide to do a water change, and you rapidly change the mineral content/TDS of the water. This can create a potentially hazardous situation for this fish, which is likely to blame for the die-off that is sometimes seen after a water change in these circumstances.