Netflix has reported some server issues today, though they seem to be recovering already.
We are aware of reported service issues and are working on a resolution as quickly as possible.
— Netflix CS (@Netflixhelps) June 20, 2016
On reddit, people say it only lasted for about an hour and is back up. We thought we were having internet issues connecting on our FireTV, but after going into Network Settings and resetting the Netflix app, it was able to connect.