Furnace Burners go off & Blower Stays On Before Setpoint Reached

Mid efficiency furnace started acting up a couple years ago. Use a Honeywell Magicstat thermostat to control. When the setpoint on the thermostat changes by more than 1 degree, the furnace comes "on" and heats up and acts as one would expect. Once the thermostat measures an increase of 1 degree C, the furnace burners shut off and the blower stays "on". Note that the temperature has not reached the setpoint. It has only gone up 1 degree. The furnace blower stays on and circulates the air, and the burners never come back "on" again. In fact, the temperature will start to drop in the house and the burners never come back on. But the blower stays on. The only way to get the temperature up to its setpoint is to manually increase the thermostat 1 degree at a time, wait for the temp. to go 1 degree, then increase it 1 more degree.
The thermostat was suspected to be the problem and has been replaced. The problem is still there with the new thermostat.
Suggestions anyone? Has anyone heard of this happening before?
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BL wrote:

Most likely it is just a coincidence that the room temperature rise is 1 degree C before the burners shut off. There is some other cause at work such as the burners shutting down on high temperature limit. Is your filter clean? Any returns and/or supply vents blocked?
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You havn`t had it serviced for Years you say, So it is about time, Yes. Nobody should guess, call a pro , it is broke.
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On Tue, 18 Jan 2005 14:06:55 -0600, snipped-for-privacy@webtv.net (m Ransley) wrote:

Sounds like Ransley's way of say, "Im on webtv and I dont have a clue" Bubba
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On 18 Jan 2005 11:14:32 -0800, snipped-for-privacy@saskpower.com (BL) wrote:

Replace the thermostat again till it does work. Bubba
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