Does ASP.net have scaling problem?
Firstly, I would say that no matter which technology, when more user and more user start using it, it will always increase decrease performance, even for the most popular search engine around the world. I am confidence that if suddenly every people start using the application together, the application will slow down. The real issue on scaling basically is what is "trigger" and by "how much".
If you design your application which can scale to 100 million users and the "trigger" happening when 50 millions user start using the keep and performance start degrade significantly after it, then we have a scaling issue.
Also, if we design an app and expect average wait time is 5 seconds, and when more people user our app and we see the expected wait time go beyond it, we have a scaling issue.
Here is a good article from MSDN magazine talking about ASP.net scaling issue.
http://msdn2.microsoft.com/en-us/magazine/cc500561.aspx
In general, ASP.net as a platform itself doesn't really have scaling issue. The issue is whether we know how to measure scaling issue and how to resolve them using the right tool provided.