Recurring intermittent problem: Furnace runs properly, but more and more often, the thermal (350 deg) switch above the leftmost burner trips, which causes the blower to stay on and the furnace to NOT re-light when the thermostat calls for another cycle. I can reset the switch (depress the small plunger) and it will work fine for a few cycles, but then trips again. I replaced the switch, but still the same problem. Any thoughts? Thanks.

If the furnace was operating properly the thermal safety device would not trip.
Usually that is a indication of to high a temperature in that area.
When was the last time the gas pressure was tested? ((You did not state what kind of fuel the furnace uses.))

To high of a firing rate will cause this problem.

If the flue pipe clear of all obstructions? A back up in the furnace exhaust/flue pipe can also cause this problem.
It is possible that the burner orifice on the burner tube right below or next to the limit might be partially restricted not allowing the burner to run a full capacity allowing some of the heat to roll out of the burner area and tripping the switch.

((You did not state what kind of fuel the furnace uses.))

>>Too high of a firing rate will cause this problem.

If the flue pipe clear of all obstructions? A back up in the furnace exhaust/flue pipe can also cause this problem. It is possible that the burner orifice on the burner tube right below or next to the limit might be partially restricted not allowing the burner to run a full capacity allowing some of the heat to roll out of the burner area and tripping the switch.>>

It's natural gas. I don't see any obstructions. I think the comment about the heat "rolling out" is viable, but I don't know what to do about it. Any suggestions? Thanks for the response.

So you can see inside the exhaust pipe that goes outside. I never could do that. I always had to look outside then fish a wire electricians tape from inside to out to be sure of no obstructions.

Since I am not there I cannot tell what might be causing you roll out situation. It will probably be something that will call for a visit from your local HVAC pros.

That may be it... something in the flue causing backpressure/excessive heat? I think I'll call the service guy. Thanks for your replies.