If it was truly any part of the timing belt system you are probably not going to like the results. Most of the small engines like that use what is known as an interference engine. What that means is that the valves will stick down enough that the piston can hit them. In a normal engine this is not a problem because the timing belt keeps the valves moving in time so the valves do not hit. But if the belt breaks or stops the valves do not move and the piston(s) can hit them. The usual result is at least rebuilding the cylinder head or replacing it.
However, your description does not really fit with it being the actual timing belt or any part of it failing. Normally when those fail the engine stops now. It will not run for a while slowly losing power until it stops. It just stops running.
What you describe could be caused if the serpentine belt came off and the engine started getting hot because the water pump was not turning, or if the power from the alternator was lost and the engine computer started shutting down.
Which engine do you have?
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Thursday, October 5th, 2017 AT 8:47 PM