I talked to a Square Enix support agent and I'll post the conversation here. This may or may not solve the issue, I haven't tried it yet, but it's worth looking into.

Agent Melinda: Hello and thank you for contacting SQUARE ENIX Customer Support. Just a few moments while I review your question, please.
You: No problem.
Agent Melinda: While I look over your ticket information, may I have your first name please?
You: ******
Agent Melinda: Hello ******. May I have the SQUARE ENIX ID?
You: ******
Agent Melinda: For the PlayStation version of FINAL FANTASY XIV: A Realm Reborn, please follow the below steps to resolve the issue.

1. Launch the Internet browser from the PS4.
2. Once the browser opens, please open the Tools Menu.
3. Delete your cookies.
4. Go to the cookies option in the Tools menu and make sure that cookies are permitted.
5. Go to "Javascript" (also in the Tools menu) and make sure that it is set to "On"
6. Launch the game and connect.
You: Alright, please give me a moment to try that.
Agent Melinda: Sure, let me know what happens.
You: I tried twice just to be sure, and it doesn't look like that had any effect.
Agent Melinda: Are you currently wired or wireless connection?
You: Wireless, however this is the first time I've experienced this particular issue in my year and a half of playing.
Agent Melinda: In order to fix the issue, we suggest you open up the direct FFXIV ports. Here are the following ports.

TCP Ports
54992~54994
55006~55007
55021~55040

For more information, please visit www.portforward.com. Also, it is insisted that you contact your Internet Service Provider to ensure that the ports are accessible to see if your connection is timing out anywhere. After you have done this, you will be able to connect to FFXIV with no issues. Also, you may want to try a hard wired connection for better results.
Agent Melinda: Do you have any questions on what you will need to try?
You: Let me check out portforward.com briefly and see if I end up having any questions pertaining to the site. Also, are there any specific questions that I may need to ask my ISP?
Agent Melinda: You can provide them the ports and they will be happy to walk you through portforwarding steps.
You: Alright. I'll see where that gets me. If I still have issues, I'll contact support again. Thanks very much for your help.
Agent Melinda: Thank you for visiting the SQUARE ENIX Support Center! Take care and hope to see you online!