OK, I've got a reply from SE:

"We are sorry to hear that you are having issues with with your connection.
The First thing to do would be to try connecting your computer to your router
directly if you are using wireless internet and ensuring that the correct
ports have been forwarded on your router. You can find out more about
forwarding ports on the site
http://portforward.com/english/applications/port_forwarding/Final_Fantasy_XIV/"

I already had my PS4 connected to my router via cable so I just did the port forwarding steps listed in the suggested article. It didn't help one bit...