No, your description indicates that it is likely a furnace burner that, for some reason, did not fire up when requested by the thermostat. Did the fan start up for a short time then quit?? The hint here is the fact that, in your description, you said it did start up normally after being turned off - then on again.
The furnace, when requested, will attempt to light 3 times, (3 small clicks of the igniter), if that fails the furnace is locked out and has to be reset by turning off the request for heat and then turning it back on. To prove this, of course, you have to catch it in action and see if the thermostat is making a request when the temp actually drops. If the fan started then stopped a minute or so later, then the request is being made and you have a furnace ignition problem.
I would connect a voltmeter to the heat request lead and see what happens. By raising the thermostat set point 4-5 degrees, you should see 12V appear on that lead when the thermostat requests heat. Which wire is that? Knowing the thermostat and furnace brands and models might help us find that.
On edit: After seeing your last post, Gary's suggestion makes the most sense in order of checking it out.