Finally I went to the Apple Store Genius Bar and explained it to them. They couldn't reproduce the issue but they kindly provided me with a brand new iPod. [Clue: they tested it OUT of the case.] Kudos to Apple.
So I got home, charged it for 3 hours as recommended, put it BACK in the case and started to listen to music. It shut off right on cue! OK, I thought, it's just a coincidence. I reset it and started playing music again. This time it worked all the way to work on the bus and then shut down when I put it in my pocket to walk up to work.
I had had a suspicion that something to do with shutting the (magnetic) case lid was to blame but had never properly tested it. So I removed the iPod Touch from the case and put it in my pocket without any case on it. And... just like that - the music kept playing all the way on the rest of my walk to work. Now we're getting somewhere!
Very frustrating! I Googled for the problem (why didn't I think of this before? Who knows!) and found somone on MacRumors who had the exact same problem! And furthermore, they had a case with a magnet for keeping the lid shut. I got one of these Cygnett cases when I bought the device and had it in there the whole time. This morning I have been testing the device with the case FAR, far away and no problems at all. I will keep testing when I walk home tonight, but I feel that it is solved. It's a shame as the case is quite nice, although now that I think about it, the fit is not great and typing on the keyboard at the edge of the screen is irritating.
BTW I did discuss this in the Apple forum, but it seems Apple still sell these types of cases in their own store!
Next step: buy a nice NON-MAGNETIC ipod touch case and screen protector from Amazon
Moral of the Story: do not buy an iPod Touch case containing a magnet!
UPDATE! (Jan 2010)
I noticed that I've been getting quite a few visitors to this post, presumably those with the same issue. I wanted to tell you that since buying one of these rubber cases
I also got a pack of screen protectors like these
I definitely recommend steering clear of the cases with magnets. While it's a shame, because the magnets are cool, it's cooler to actually be able to listen to music properly!
BTW, if by the time you read this, the case above is not available, I suggest looking here to find more