In some environments where Clean URLs will in fact work,
the Clean URLs test shows the page with the test button and no tick-box
(giving the false impression of a failed test result)
if called from
but gives a Clean URLs enabling page with tick-box (the correct result) if called from
"http://www.example.com/admin/config/search/clean-urls" (note clean address).
I think the affected environments must be fairly rare in the overall scheme of things, otherwise it would be a more famous bug. But I don't know exactly what the critical difference is.
I encountered it when using D7.0 in conjunction with an "add-on domain" (method of cut-price hosting, involving a sort of redirect). A non-addon-domain trial install similar in most other ways did not show the bug. See discussion at http://drupal.org/node/1167698. Any questions not already answered there, feel free to ask.
(I've only just seen that there's a D7.2 now, and haven't yet done my upgrade. I don't see anything relating to Clean URLs in the notes for 7.2, so I think that means it will be the same. But that's yet to be tested.)
I was eventually clued in to the solution by others' previous experience as reported at http://drupal.org/getting-started/clean-urls#comment-2526658 and http://drupal.org/getting-started/clean-urls#comment-4511508. (The earlier of those referred to D6, the later to D7.)
Other comments further down that thread, http://drupal.org/getting-started/clean-urls#comment-4153652 and http://drupal.org/getting-started/clean-urls#comment-4222904, look to me like they might also be the same thing. I say that because the commenter reports a clean address actually working even though the test says no.
http://drupal.org/getting-started/clean-urls#comment-4007886 reports a similar thing but with the added factor of overlays. In my case I had already ditched the overlay and was still getting false negatives with the unclean direct address.
I'm setting this to major because of the "significant repercussions" to the few people who are affected - a wrong test result is very very misleading and time-consuming! But an interim fix would simply be to warn people to try the alternative address before believing the test result. I'd suggest adding it high up on the handbook page immediately, and on the settings page in future updates (unless the bug can be tracked down and fixed straight away).
Extensive discussion of the Clean URLs settings page already exists at http://drupal.org/node/881376 and http://drupal.org/node/423196, including mention of a false negative tentatively attributed to a clash with overlays. (I've copied the "component" label "base system" from one of those as I couldn't find any component in the drop-down list actually called "Clean URLs".) But neither of those threads is specifically about this particular bug, and I'm not finding it elsewhere in the issue queue either, hence creating new issue. Hope that was the right thing to do.