Thanks for this thread. It helped me to quickly resolve my issue. I will elaborate on my own situation.
My site's Moodledata had permissions 770. I set it to 775 and the problem was resolved.
This seemed particularly odd to me, as the folder is owned by apache:apache, and the moodledata folder should be (is) only accessed by 'apache'.
I saw that at least one child folder was owned by root... I ran a 'sudo chown 775 moodledata -R' on the folder, and the problem was then solved - functioning quizes with only 770 access.
I'm a little baffled by this, as I had previously chowned the whole folder to apache:apache. I'm not sure what could be changing some of the contents to root ownership? Maybe I had a sysadmin space cadet day...
tl;dr - check that there aren't some subfolders/files in your moodledata that are owned by someone else