Yes and no. What happens on some is the attenuator if equipped gets burnt and will not allow the current to flow. This is either a bar or a coil that allows you to adjust how close to set temperature the furnace runs. If equipped it is normally set about .80. You can slide the bar or turn the setting up or down slightly and the furnace will run.
Another problem is with the contacts. They get burnt and will not conduct. You could use an Emery board and clean them a little and they will still work.
Yes, the wires are small and somewhat delicate.
You can get a replacement mechanical for around $15.