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Thanks for the replies, Ehel and AC. I have secured the router, so that won't be the problem, I don't think. Unfortunately, turning the router off at the plug simply has the effect of turning it off at the button!
When the connection fails, it's on every webpage as well as other things that don't use the same browser. The ISP is Tiscali.
As for the other advice - what happens when you press the button sometimes is that the lights don't actually light up, they glow dully. Leaving it to do its thing when they're doing that has no effect whatsoever.
Pinging the router when it's in this situtation leads to failure, so I'm assuming it's the router rather than anything else which has the problem.
Any further suggestions?