After installing the latest Steam client (http://store.steampowered.com/) on my Ubuntu 14.04 using the Ubuntu Software Center, I decided to install another package using the package manager.
As a first step I ran apt-get update, which (you guessed correctly) failed fetching several index files:
Err https://private-ppa.launchpad.net precise/main amd64 Packages
Err https://private-ppa.launchpad.net precise/main i386 Packages
Ign https://private-ppa.launchpad.net precise/main Translation-en_US
Ign http://security.ubuntu.com trusty-security/main Translation-en_US
W: Failed to fetch https://private-ppa.launchpad.net/commercial-ppa-uploaders/steam/ubuntu/dists/precise/main/binary-amd64/Packages HttpError401
W: Failed to fetch https://private-ppa.launchpad.net/commercial-ppa-uploaders/steam/ubuntu/dists/precise/main/binary-i386/Packages HttpError401
E: Some index files failed to download. They have been ignored, or old ones used instead.
Searching for a possible reason led me to Artūras Norkus (xeranas)'s answer: https://bugs.launchpad.net/ubuntu/+source/software-center/+bug/1130945/comments/4
It seems that my /etc/apt/auth.conf pointed to an incorrect username (a user that was used to merge between 2 different accounts that I had), changing the username to my current username fixed the problem and I was able to update the index files with no errors/warnings.