@Reine Marge - My host has informed me there is a CPanel issue with WordPress file ownership. I will try to explain this how I understand it, but if I am incorrect, someone please advise:
At times, file ownership is assigned to "Nobody" within Cpanel (not sure if this is for Fantastico installs only or if manual installs have this too). So when you run the core update, you do not have permission since "Nobody" is assigned permission.
The solution has to do with the hosting server setup (suPHP and PHPsuExec), which I do not have the expertise to explain.
It's frustrating, and from what I am told, not an easy fix. Hope someone out there might have a solution :)