So far I have been using Twisted to simultaneously serve a lot of mobile clients (Android, iPhone) with their HTTP requests exchanging JSON messages.
For my next project I'd like to try out Google App Engine, but I'm wondering if it is capable of doing the same or if I should rather go with a custom built solution again.
Certainly. App Engine will scale your application up as the load increases automatically and will be spread over many machines. The web api they have is pretty nice too. You don't have to worry about deferreds either because it scales by bringing more instances up instead of making things asynchronous.
BTW: I have web services hosted on app engine that are consumed by iPhone.