A pair of issues have cropped up on North American servers and Game Director Yoshida has gone on record via the Final Fantasy XIV forums to let players know that the team is aware of and working to solve them. The problems are being caused by high stress load and congestion.
First, certain instanced battle content is overloaded. To address this, 20,000 playable slots have been opened up to players of instanced content each hour.
Secondly, Duty servers are "experiencing continuous crashes" due to "consistently high stress loads" due to the number of matching requests from data center Worlds. Players are receiving a 90006/90002 error and the team is trying to track down the root cause. A new work flow has been enabled to restore the worlds every 15 to 30 minutes if the Duty server has crashed.
You can check up on the progress and participate in the discussion on the Final Fantasy XIV forum.