There's no reason to even think about that possibility. One of the first things you'll want to check is the hose that goes from the lower edge of the outer tub to the diaphragm on the fill level selector valve. If it's cracked or hardened with age and no longer sealing properly on the diaphragm fitting, the captured air pressure will leak out. Depending on the size of the leak this can happen slowly or quickly. Since that air pressure is the only way your machine knows how much water is in the tub, if it's a quick leak, the machine will simply never stop filling and overflow. If it's a slow leak, the machine will appear to be functioning normally. It'll fill the tub with water and begin agitating. But once enough air leaks out it'll think the water level is low, stop whatever else it was doing, resume filling and overflow. You'll probably have to tip the machines top panel up and prop it open to have a look. I have an A806, and the set up is a bit different, so perhaps someone here that has an A906 can help you there. At any rate, this is so easy to either confirm or rule out, that I'd check this first.

It does sound like your fill valve stuck, as Martin suggested it is fairly easy to replace this valve [ some can even be rebuilt it depends which one your machine came with ] . This is a potential problem on any older DW or washer and I would advise ANYONE with a DW or automatic washer that is over approx 30 years old be replace or rebuild the inlet valves in there machines or DO NOT leave then unattended when they are running and TURNOFF the incoming water supply when you are not using the machine.