I had a problem with updating my WordPress 3.5 to 3.8.1. I simply got the following error message:
Unpacking the core update Could not copy files Installation Failed
The problem was with the file ownerships, and the fix was found at http://codex.wordpress.org/Updating_WordPress
All of your WordPress files must be owned by the user under which your web server executes. In Ubuntu, the Apache web server user is www-data, so I simply had to run a command:
/var/www$ sudo chown -R www-data rusama.net/
And Voilá, everything started working smoothly!