My first guess was that the flow sensor was wonky and it had trouble detecting water was still being used. As it was the warm season and the heater was still on the winter setting (hottest water), the hot water flow was minimal. I tried adjusting the heater for less heat and recalibrating the flow sensor, but no dice.
So I went to the vendor's web site to see if there was a local service outfit that could come and fix it. Part of the appeal of this heater is that it's designed to last a lifetime, and everything on it is replaceable. When I first bought it, there were no local contractors, and I took a bath on the installation as the plumbers I hired were not familiar with it. The first outfit I called didn't have any openings for three weeks. The second outfit was very polite and responsive and sent a guy right out. He watched the heater run for a while then shut down and announced he had no idea what was wrong with it, didn't want to swap parts randomly, and suggested I just buy a new unit. As it was a replacement, installation would only be $100, but the new heater (with a 15 year warranty) was $1600. After some dithering, I decided to go ahead with it. They didn't have any in stock, so they'd have to order one and install it when it arrived, in a couple of days.
As the days passed, the existing unit worked less and less well and we were getting truly sick of cold showers, so I started to tinker with it, as there wasn't much worse I could make things. I realized the problem was with the flame sensor, as the heater would be running merrily, then the ignitor would start sparking, the pilot flame would come on, the main burner would shut off, the sparker would keep sparking, then the whole thing would shut down in a funk.
The flame sensor appeared to be a simple L-shaped bit of heavy wire that stuck in to the flame. I played with it a little, and the heater would work briefly then quit again. Then it would refuse to restart for a while.
I realized the wire from the flame sensor was a little loose, so I recrimped it, and it ran nicely. Then it quit again. I figured perhaps it was a grounding issue, and got worse when things warmed up. So I took apart the ignitor assembly, cleaned everything, and reassembled it firmly. Ran nicely for a while, then quit.
Then I decided the flame sensor itself might be flaky (it's red hot when the heater is in operation). I didn't know how it worked, but the wire was a single conductor. Many gas appliances use a thermocouple or thermopile as a flame sensor, with a single wire connection (the ground return being through the burner). So I went shopping and found a generic replacement thermocouple and wired it in. No dice.
More research revealed that modern equipment uses the electrical properties of flame (which is conductive due to ionization) for flame detection, and this is a very safe method as the usual problems (shorts, opens, resistance) don't have the same properties as actual fire. This is known as "flame rectification". I tried hooking up a diode in series with the flame rod and the heater worked! Temporarily.
So I went to a heating supply house and bought a random flame rod that looked approximately like the one in the heater (it turns out there's no mechanism, just a little Kanthal wire and an insulator). This one was longer than the original, so I made a bracket to hold it at the proper place. The heater fired up and worked and I had a nice hot shower! Unfortunately, fizzygeek couldn't get any hot water at all for hers.
I had ordered the correct flame rod and also a replacement ignitor module with overnight shipping, so they showed up that afternoon. I put 'em in that night, and the heater works perfectly again. I called up the plumbers and told them that I would not be needing the replacement water heater after all, saving myself a cool $1500 or so.