Netflix says 'You have downloads on too many devices. (10016-23000)'

If you see an error on your iPhone, iPad, or iPod touch that says

You have downloads on too many devices. Please remove downloads from another device, or go to netflix.com/changeplan for plan options. (10016-23000)

It means you've downloaded TV shows and movies on as many devices as your plan allows. To download on another device, you can: