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 […]