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The first thing you should try is resetting your router by turning it off and then back on a few minutes later.

Make sure that any anti-virus or firewall you have has an exception for Final Fantasy XIV (or disable it temporarily while you download the patch).

Check the following Port Ranges on your router/modem to ensure that the following Port Ranges have been opened/forwarded/triggered:
▼TCP
54992 through 54994,
55006 through 55007,
55021 through 55040
*This can be done by visiting www.portforward.com, locate your specific Router/Modem Brand and Model Number from the 'List of Routers', and you will find a step-by-step guide on how to access your ports.

We hope we can be of assistance in helping you enjoy the exciting world of Final Fantasy XIV!