I was cycling along, and then I heard a gentle "sh ... sh ... sh" noise coming from the rear wheel, as if something was caught in it. This noise became more frequent when I went faster, even if I wasn't pedalling. I stopped to take a look, but I couldn't see anything. However, when I tried rolling the bike forward (to watch the rear wheel) it stopped, and I had to give it a bit of a push to keep moving.
I was only a few minutes away from my destination at that point, so I just kept going at a gentle pace. When I arrived, I tried to fold up the bike, but I couldn't move the pedals round (in the normal direction) to get the right pedal out of the way of the front wheel. I had a closer look at the rear wheel, and discovered the problem:
If you look at the rim next to the valve, you can see that it's a bit different to the surrounding area. Here's a close-up:
Basically, it's cracked in the middle, so part of it has bulged out and it won't fit past the brake pad. By contrast, here's what the rim is supposed to look like:
There's a black line that runs through the middle of the metal surface, which is actually a groove. Basically, rim brakes (as opposed to disk brakes) work by friction: you're squeezing them against the rim, to bring the wheel to a halt. This process gradually wears away the rubber on the pads, so you need to replace them every so often. However, the rim also wears away. As that happens, the groove will get shallower, until it eventually disappears completely. (This is the same principle as the tread depth indicator on car tyres.) On the broken rim, the surrounding area is completely smooth, i.e. the groove has disappeared. Eventually the force of braking was enough to crack the rim, and within five minutes the bike was completely unrideable.
Looking through the owner's manual, there's some relevant information in section K, item 9: "Wheel rims: when either groove in the braking surfaces is no longer visible, the rim should be replaced." (It says "either groove" because there's one on each side of the rim.)
I've probably read that before, but I must have just skimmed over it. I've been cycling since I was a kid, but I think this is the first time I've used a bike for long enough to actually wear down the rims, so I'm continuing my "trial and error" approach to bike maintenance. (In the past, I either outgrew bikes or they were stolen.) I didn't track my daily distances for the first seven months that I owned this bike, but I know that this rim has lasted at least 8000 km (5000 miles). It may be significant that the rim cracked next to the valve, if that's where the inner tube exerts the most pressure against the rim.
Although you can see the groove, I think it's best to feel it. Run your finger across the rim (from the tyre towards the spokes) with your fingernail against the metal. When you reach the groove, you will feel your nail catch in it. This won't hurt, as long as you move your finger gently. This will give you an idea of how deep the groove is, rather than just "is it there or not?" I think it's best to do this whenever you check the pressure on your tyres, since you'll be looking at the wheels anyway and it only takes a few extra seconds.
When the rim wears out, you will need to replace it. The simplest approach would be to replace the entire wheel, but you may want to keep the hub, e.g. my front hub is a dynamo and my rear hub has the gears. (Spokes aren't particularly expensive, so you may as well replace them at the same time as the rim.) In that case someone will need to rebuild the wheel; I wouldn't feel confident about doing that myself, so you'll be without the bike for a couple of days when it's in the shop. The advantage of checking the groove regularly is that you can plan around this, and you won't get stranded in the middle of nowhere.
I've checked the rim on my front wheel, and that's ok; the groove is shallower than the new one at the rear (as I'd expect) but it will still keep me going for a while yet. This implies that I've been putting more force on the rear brake. Sheldon Brown has some advice on this topic, saying that it's best to use the front brake most of the time, so I'm now making an effort to change my habits.