In a Shared environment, if the file owner is apache:apache, nobody:nobody, root:root, or any combination of those 3 users, you must have root access to change it.
SSH, cPanel/DirectAdmin, WHM, whatever, but it must be the root account.
I had this problem before I got my dedi. And to my Host's credit, he was very prompt in resetting permissions for me when I requested it back then.
(And once I enabled suPHP, the occasional hiccups with file ownership stopped, as well as relieving me of the need to change certain directories to 777.)