If you are using a router, make sure you have port forwarding set up whenever you are downloading the patches. It uses a torrent system and it is quite possible your router is not handling UPnP properly.

Use the following settings in your router:
Start ~ End Port --> Protocol
55296 to 55551 --> UDP
54992 to 55551 --> TCP
443 --> UDP

Also, make sure you have Windows Firewall not set to block it. With all that, it should be fine. If all else fails, go to Square's Final Fantasy XIV Support page.