Then if Server A blows up, I simply ask my ISP to route the packets intended for Server A to Server B. (I assume they can do this if it's their backbone)."

Another approach to this would be to use a content distribution network (CDN) that has application and database caching functionality baked into it. Take Akamai for instance. Their EdgeSuite service allows you to cache record sets or entire databases on their network, make updates to the cached data, and update the master database on your server. If you make changes to the database on your server, you can push a fresh data set to the Akamai network. Similarly with your web applications, you can program them to run on the Akamai network. This combination of application and database functionality reduces your infrastructure requirements to a single server and the Akamai service. All you need is a server to act as the authoritative record for your application and data. If your authoritative server blows up, your application keeps running on the Akamai network.

However, there is a trade-off in that you must program your web applications to use ESI (EdgeSide Includes), Akamai's scripting language that runs on their network. For more info, check out Akamai's developer site.

I have yet to see any other CDN service provider offer such an wide and flexible array of services and functionality. I'd welcome experiences with other CDN's.