This is a good question. I suspect not. Servers crash because too many people want to go on at once, which means there isn't enough computing power left to send you your video. The link below outlines two ways that HBO get around that.
First, HBO rents their servers from Amazon Web Services (aka AWS). This means they have "elastic computing", which means that if they run out of server capacity, they automatically rent some more. AWS's business isn't just to power HBO, but the whole internet, so it's unlikely they'll run out. They don't just have one data centre full of computers, they have multiple data centres in most major cities in the world. If the data centre in your town stops working, you'll be re-routed to another one.
Second, HBO uses Kubernetes, which is software that watches how much computing power is being used and rents more servers early. (The word "kubernetes" comes from the same Greek word that gives us "governor"). That way, you shouldn't have to worry.
Could it break down anyway? Yes. One problem is that there's no gradual ramp-up of users. There's just a sudden spike of watchers at once, which doesn't allow kubernetes any lead time to start spinning up more servers. You just have to wait for Amazon's Elastic Compute to boot up more servers, which can take seconds.
But to get around that, they do stress tests beforehand. For instance, in the months leading up to the season launch, they would have done simulations. They might simulate several million users, to see what happens to the system if they all try to connect at once. They might simulate shutting down one of the data centres in the middle of the episode, to see if Elastic Compute can add more in time to prevent the GoT being laggy while you're watching it.
But if it really does crash, the crash should be temporary. If all the AWS servers in all the cities go down at once, we'll have a lot more to worry about than missing our favourite TV show. Hope that helps.
https://www.eweek.com/cloud/why-hbo-chose-kubernetes-to-help-stream-game-of-thrones