We should hope they are using AWS infrastructure.. That being said, if they are, they don't appear to have provisioned autoscaling very well either for either the game servers (or website for that matter), or have placed some arbitrary (read "financially driven") limit on it. Certainly AWS has the capacity, globally, to deliver this service.
Amazon has the capacity, when it is not christmas season. Every year before christmas Amazon sales go off the scale, that is why Amazon developed their AWS, in the first place.
Now, what is your guess happens if Amazon has to decide which servers have priority?
...and no, they won't get in trouble with customers, as the usual contract has something like 99.5% uptime guaranteed, so the odd 20-30 minutes every day around christmas (distributed over 4 to 6 5 minutes server downs) are well inside the Service Level Agreement.
Unfortunately, it is quite annoying if you play a game and not just loading a website and just have to click a reload button for it to work...