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I have now managed to fix this problem which occurred again on another page.
Luckily, this page was a "copy" of another page which still worked, so all I had to do was compare the two pages. I discovered the difference was that they were using subtly different page templates.
I realised it was the old #FORM_OPEN#, #FORM_CLOSE# problem, and I changed the two #FORM_X# locations in the faulty template to be "tighter" around the #REGION_X#s. I had previously put the FORMs around all my #REGION_#s, even non-input ones, but maybe I had an unmatching div or something, and it couldn't find the FORM_CLOSE.
So now it works!
Sorry, I can't put up the faulty template, because I can't remember where the FORMs were when it was broken (and I'm a bit of an amateur at page templates, so they look really bad)!