jkbdavis said:
Is there a sensor on the furnace itself which allows the fan to turn on?
The thermostat is likely wired to a controller that is responsible for signaling ignition. That controller might be jacked up.
I had a prob a couple months ago:
There is a flame sensor which can shut the furnace off if the furnace hasn't properly flamed on a few seconds after the power switch opens the valve for gas. Sometimes, when the flame sensor gets oxidized, it gives a false alarm; it can't see the flame, so the furnace could cycle: thermostat triggers gas, flame sensor shuts gas. thermostat sees it's still cold, triggers gas. Flame sensor shuts gas. So on and so on.
My fix was to sand clean the flame sensor. But sometimes unexpected stops like this can jack up the electronics that control the furnace. Try powering down the furnace and then restarting and see if yr problem goes away...