See if you can successfully ping the lobby service. For Aether, it is neolobby02.ffiv.com. Primal is neolobby04.ffxiv.com. Chaos is neolobby06.ffxiv.com

If you aren't familiar with ping, open your search/run box and search for the letters "cmd". This program opens a command prompt, where you can run some testing tools. You should have a blinking cursor where you can type commands (if you have lost focus on the window, it won't show any thing you type...just click anywhere in the window to get back to the input line). Type the following phrase, editing it for where you want to test connectivity to, and then press the <ENTER> key (can use a web address, IP address, or registered server name as used in this example):
Code:
ping neolobby02.ffxiv.com
If it responds to ping requests, then you can connect to it. If it times out (* * * result), there is a network/security issue to look for.