I got the same problem yesterday, you can check the chmod for folders i gives 0777 to make it work fine, i`m using cuteftp and its really easy to change the chmod, or u can do it from shell :
i tried this, but the problem is the same. i'm not shure if there is a problem with the owner. i asked the hosting company to change the owner of the complette installation to apache.
Check, that you can upload with PHP (but I guess, you did this check with other PHP scripts).
Then check the rights.
If you "chmod" them, then do it recursivly in "var". Maybe your installation has a temporary folder, where uploaded files are places... If so, check the file rights there, too.
Silly question: Do you have enough space? ;) Try with small files.
Is the problem with files or with images, or both? If images, check the hints in "documentation" for ImageMagick config (image.ini.append)