We are talking a District Line here with its constant stops between stops and delays, but very few people managed to beat the Tube. However, Alantan found a video where a group of students tried a similar idea with a SPT subway train in Scotland
"You are on the Glasgow underground (subway), travelling clock-wise. The journey time between Buchanan St. and St. Enoch is approximately 55 seconds. Hmmm. On the surface it's a downhill journey, down the busiest shopping street in Glasgow with 2 road crossings. Can you get off a train at Buchanan St. and back on the same train the next time it stops at St. Enoch?"
Well it's not totally man against subway, but watch the video (wmv or mpeg4 - the wmv only seems to work for me in IE) to find out if they did it. I particularly like the lazy Jamie Cullum soundtrack as the subway is ambling along, compared to the frantic music when the student is racing against it outside.
Back to the Beat the Tube Race, the author of Eccentric London reckons they would stood more of a chance cutting across "the Circle Line from Farringdon down to Blackfriars where you can virtually see the other end. You could do it on crutches, or pogo sticks and still beat the train".
Not that I'm suggesting that you try to outrun trains on a regular basis, but are there any other parts of the London Underground network where you think you would be able leave a train, leg it and then catch the same train again?