I got a reply back from someone at Dell helpfully telling me to "use a paperclip or similar device to press and hold the reset button for 5 seconds". This was a very odd email to get as there is no reset button of any kind on a Latitude C400, with or without a paperclip.
At this point I decided that I would brave the Dell phone lines -- something that I've in the past sacrificed whole working days to. However, fairly fast I got a human who, when I told him my service tag number, suddenly realised that I had signed up for their "Gold" care, and put me through to someone who could "actually help me". I told the new guy what was going on, he got me to describe exactly what happened when I turned on the power button, told me that the memory had gone on the blink, and that he'd send someone to come and repair it on Wednesday. I shall report back.