Hmm, now that you mention it, APC ( http://php.net/manual/en/book.apc.php
) is installed and enabled on this server.
I bet that's the problem.
APC was installed to enable upload progress bar functionality.
Now that we know that the problem is related to APC, it was easy to find people reporting similar problems:
Maybe this is simply a bug, or maybe APC can be configured to allow for this kind of "building" behavior.
Thanks for the help, Till!