In case anyone has this problem, it appears to have been the battery. Lenovo sent me a new one and I've been using the laptop for 2 weeks now and it hasn't shut down a single time. Must have been something wrong with the circuitry of the brand new battery that caused it to suddenly cut out at times. Not real impressed with Lenovo's quality control there but I am happy with the x220t now that it works .
Ok, I have an X61 and it shows pretty much the same symptoms. It shuts off suddenly and completely---all lights off---when running on batteries (even when in sleep mode) and will not come on again until I have removed the battery (and it must also be disconnected from the mains to come on!). I have now replaced the battery. I shall report here again when / if I have the solution.