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Additional information:
1) The furnace usually has a low hum, like a transformer. But when it stops working the humming stops. Flicking the switch starts the humming and starts the furnace. It's like the furnace shuts itself off.
2) This problem happended again last night - the third time in two weeks. It's always at night and always when the outside temperature is under 5 degrees. I live in Rochester, NY. I wouldn't think it would be possible, but could the exhaust be condensing in the chimney, freezing, and constricting the outflow of air? The chimney is regular stainless steel with about 5 feet of pipe extending above the top of the house. There is a stainless steel cap on top.

matter