I have just checked. They can. Any ideas?
Well, you could have routed it manually (by accident) and you could have blocked it with a firewall.
In theory we could next check and double-check if you can access linuxmint.com when you boot that computer to a live instance (USB o DVD), just to assure there's nothing going on at the router level specific with your IP address or MAC address - but that's relatively unlikely.
So, do you have a hosts file that you've manually edited? Are you blocking the software with some sort of firewall or iptables rules?
You've pretty much eliminated everything else - so we can safely conclude (more or less) that it's just with that system. I mean, you could check a live instance but most folks are gonna remember if they set up rules on their router. That's not something one does by accident - unless you drink a lot of wine?