Are you using a wireless connection, or hardwired with an ethernet cable? If your answer is wireless, what quality would you say your router is? I have a mid-range router (about $100 range) and I had the same type of problem happening when I was playing Warframe. I wasn't able to connect to the servers of the game, and it kept telling me there was a PSN problem, even though I was logged into PSN and could see my online friends list and everything. Was even in a party chat with my brother.

While I was raging my bro looked up stuff about it and said he found an article saying that the PS4's wireless is very taxing on routers, and to have optimum performance you would need a top of the line router. He started listing routers that costed $250 and up. I said screw that. I went and bought a 25' ethernet cord and hardwired my PS4. I haven't had a single connectivity problem since. I haven't ever lagged once in FFXIV on PS4. I've had a few minor framerate drops in Mor Dhona when there is hundreds of ppl there, but no lag at all in instances and no problems logging into the game.