The most likely point of failure is with the bi-metal thermostat control. It, unlike the thermostat on your wall which activates a relay in your furnace, carries the actual current used by the heater. They overheat and/or the bi-metal sensor fails. I have no source for them, have been looking for a long time. Might not be repairable because of that..
howtomendit.com for free repair help, information and advice.
Try fixfault.com to fix oil filled radiators.