Final Fantasy XIV: A Realm Reborn Producer and Director Yoshida here: The number of concurrent connections is more than 140,000 at the moment. Congestion has also stabilized. To ensure comfortable play for everyone we’ll continue to monitor the situation. In case of problems we will keep you updated via the official forum, Twitter and the like.
140,000 concurrent connections is a very impressive number for a game in early access, especially considering that the tweet above was posted about an hour after the opening of the servers, between 3 and 5 AM in the US, depending on timezone, meaning that a sizable number of North American players were fast asleep.
Even in our comment section some wondered how many of those that took part in the free open beta would actually buy the game. Turns out that many did, as the open beta reached the record amount of 150,000 concurrent connections. The fact that early access (open only to those that preordered the game or had a Legacy account) got very near to that number is definitely a good sign.
Of course heavy traffic comes with its own set of problems, and Square Enix has just announced an emergency maintenance to address them.