Carefully examine the thermocouple tip. This is the exposed welded joint at the end of the thermocouple, which is not covered by the ceramic tube (as seen on older closed-end tubes).
To do this, you will need to remove the thermocouple from its protection tube (if it is a kiln equipped with one of our older protection tubes). You can do this with the kiln disconnected from power.
On the newer open-end tubes, you can simply look inside the tube and examine the thermocouple tip.
Look for corrosion - especially if it is severe. These thermocouple tips will oxidize and otherwise corrode over time. That is normal. There is, however, a point at which the corrosion affects the ability of the tip to function (thermocouples work by generating a small voltage at the tip caused by two different metals reacting to each other).
Ensure the two wires are securely connected. One of the factors that can cause intermittent problems is a poor weld. If the two wires touch each other (even if they are not welded) they may work temporarily. However, if the weld is not secure, then the wires could separate when the kiln heats up, causing an intermittent failure.
If the thermocouple tip looks healthy then test the control board.