I recently experienced something on my fixed - a clunk when putting down the power to accelerate from standing, which felt like the chain was skipping.
On checking the chain, it was reasonably tight, and didn't look as if it could ride up anywhere near enough to skip.
I spent a lot of time with the bike upside down turning the cranks trying to reproduce it, but to no avail. I then tried manually derailling the chain by pushing it sideways with a pair of pliers (I know not to put my fingers too near a moving fixed drive chain!) but found nothing at first, eventually, though, upon applying enough sideways force, it did make exactly the same king of clunk that I'd experienced on the road, and the chain went slack. I thought at first that the wheel had slipped in the dropouts, and slackened it off and retensioned.
Turned out the BB was shot - it seems that occasionally, the drive side bearing goes asymmetric (the only way I can describe it), making the BB on that side somewhat eccentric. My theory is that perhaps one of the balls or ball races has disintegrated somewhat, and this allows the spindle to shift. This explains why sometimes, after experiencing a slack chain on a ride, I've gone later to re-tension and found it perfectly tight.
I guess one possibility is that this has been caused by excessive chain tension (although I'm careful to check that the chain is nowhere near binding, and has a tiny bit of up and down play when I tension it) but I think the model supplied as stock with this bike has a reputation for dying early (this one has done 5,000 all-weather miles), so I'm going to replace it with a shiny new UN55, just as soon as I've found the time to remove the old one to find out the appropriate size.