Linux Mint: “Could not download all repository indexes”

When running

sudo apt-get update

or attempting to update via the Update Manager in Linux Mint (13), I was fronted with the following error:

Failed to fetch http://packages.linuxmint.com/dists/deb ... tion-en_US Unable to connect to packages.linuxmint.com:http:
Failed to fetch http://packages.linuxmint.com/dists/deb ... slation-en Unable to connect to packages.linuxmint.com:http:
Failed to fetch http://packages.linuxmint.com/dists/deb ... tion-en_US Unable to connect to packages.linuxmint.com:http:
Failed to fetch http://packages.linuxmint.com/dists/deb ... slation-en Unable to connect to packages.linuxmint.com:http:
Failed to fetch http://packages.linuxmint.com/dists/deb ... tion-en_US Unable to connect to packages.linuxmint.com:http:
Failed to fetch http://packages.linuxmint.com/dists/deb ... slation-en Unable to connect to packages.linuxmint.com:http:
Some index files failed to download. They have been ignored, or old ones used instead.

This is caused by missing repositories which are either old, broken or currently down. In order to fix it, simply run:

sudo apt-get update fix-missing

If this option doesn’t work, try changing the repository mirrors with the following guide.