If you're on Ubuntu, and all of the above has failed you, AppArmor is most likely to blame.
Here is a good guide how to fix it: https://www.digitalocean.com/community/tutorials/how-to-create-an-apparmor-profile-for-nginx-on-ubuntu-14-04
Long story short:
vi /etc/apparmor.d/nginx
Or
sudo aa-complain nginx
sudo service nginx restart
See everything working nicely... then
sudo aa-logprof
I still had problems with Nginx not being able to read error.log, even though it had all the permissions possible, including in Apparomor. I'm guessing it's got something to do with the order of the entries, or some interaction with Passenger or PHP-Fpm... I've run out of time to troubleshoot this and have gone back to Apache for now. (Apache performs much better too FYI.)
AppArmor just lets Nginx do whatever it wants if you just remove the profile:
rm /etc/apparmor.d/nginx
service apparmor reload
Shockingly, but hardly surprising, a lot of posts on fixing Nginx errors resorts to completely disabling SELinux or removing AppArmor. That's a bad idea because you lose protection from a whole lot of software. Just removing the Nginx profile is a better way to troubleshoot your config files. Once you know that the problem isn't in your Nginx config files, you can take the time to create a proper AppArmor profile.
Without an AppArmor profile, especially if you run something like Passenger too, I give your server about a month to get backdoored.