Halfway the evening, the halogens went off. Checking with a power seeker (a small screwdriver with a light in the tip that lights up if you touch a live wire) confirmed that the dimmer didn't work anymore.
So we did some calculations. I checked the packaging of the halogen spots, and they're rated at 20W. With 14 spots, that's 280W -- and the dimmer was rated for 300W. But because the dimmer was placed in a wall, we'd have to deduct 10% of the maximum power, so at 300W that was cutting it awfully close.
The dimmer was a (relatively) cheap no-brand thing, so that was the most likely cause.
Yesterday, we went into the city centre and asked at a lighting store for advice. We got a high-quality 400W dimmer -- which should be more than enough for our purpose. Back home, I installed it, and it worked quite well.
But during the evening, the halogens went out again, and the dimmer felt quite warm to the touch!
So we re-calculated, and drew a schematic of the situation, to rule out shorts, ground-leaks, etcetera. And then I checked the lamps.
It turns out that the 12V halogens we use with the transformers are rated for 20W. The halogens we use in the extension are for 230V, and are rated at 50W. It's a small wonder the fuses of the dimmers lasted as long as they did!
So I installed a normal switch instead of a dimmer -- we're used to running the lights full-on anyway. That should fix it.
Also, as an experiment, we bought a single LED spot in a halogen-like housing. It's quite bright (though not as bright as the halogens, but at less than 7W one can't expect too much, I guess). But the LEDs in the fixture are bright white, and not warm white. This gives a very 'cold' feel to it, which means we're going to give this a miss -- but there are spots with warm white LEDs, though our DIY shop doesn't stock them. Maybe we'll order a few and gradually replace the halogens with LEDs. This needs some more experimentation.
Also, in completely unrelated news, I used my birthday money to get a beer-brewing starter kit -- along with two ready-to-brew kits too. We'll see how that goes.