I've got an Amana GHI70A30 unit, it's getting old, I think about 19 now. It came with the house I bought in 2005. I had this exact same problem at the start of the heating season last year. I don't recall this problem the year before.
The furnace fires up, runs for about 20-25 minutes, then cycles off and on, and will run for about 5 minutes and repeats. If I let it sit a while, it will run for 20 again, and then begins the cycling.
The fan limit is steady throughout and not hitting the upper limit.
I've got a chimney cap on the stack, put it in 2005 and again, didn't notice issues in two years ago.
Thermostat is a programmable unit, the it seems to ordinarily work fine. AC works fine, when the furnace hits its set point, it turns off and turns back on appropriately. The problem is discovered when I set the thermo to go down at night, and when it kicks in the morning, resulting in extended running.
Tech came out last fall, couldn't find anything specifically wrong suggested replacing the exhaust blower. The blower is functioning, but he speculated perhaps it's not blowing as well as it should. When I flip the manual override to kill the unit, and turn it back up, the exhaust blower fires up and the limit thermometer drops like a rock. If you can't tell, I am a little skeptical of the diagnosis. To me, it seemed like "lets spend $600+ to see what happens." Anyway, since I paid this tech for a service call and I could find no one else to diagnose for free, rather than throw good money after bad, I said to go forward with the repair and hoped for the best. While we waited for the part, the furnace stopped doing the cycling and told them to forget it. The unit ran fine the rest of the winter... until now.