comments closed but shouldn't be

I have the general setting to allow comments, and individual posts all have the comments box checked, but some (not all) posts won’t allow them anyway. It only displays the “comments closed” message, and there doesn’t seem to be any pattern to when it decides to close them.

I tried unchecking the box for a post and then re-checking it but that didn’t help. I also tried different themes, including the default, and that didn’t help either. So it isn’t a theme problem, and the persistence of the checkbox tells me it isn’t a database problem.

I’m baffled, so any help would be very much appreciated. o.O