Hi enrewen,
First and foremost do take a look at this link:
http://codex.wordpress.org/Changing_File_Permissions
Because a lot of file upload issues seem to center around permissions issues.
That said, when you say "It's only on some of the files." Which files work and which ones do not work? Thinking file types (e.g., JPG, PNG, GIF, etc.) and size (e.g., 1K, 500K, 1MB, etc.), let's look for a pattern of what fails and succeeds. Moreover, what error message(s) are you getting when something fails? And, what operating system, browser, browser version are you using when something passes vs fails. All this kinda stuff will ultimately put the puzzle together for you, me and others who come by to help out. :-)
P.S., Might I also suggest calling your web host and letting them know what's going on. Because server folks can sometimes help out. Especially if it ends up being a permissions issue.
Hope something here helps!
Sincerely,
Randy