Right, but a third-party app that talks to both Twitter and Dropbox can integrate them both easily enough.
As far as liking third party clients goes, that's the sort of thing I'm getting at when I say that App.net doesn't solve what I see as the root problem with Twitter. Not liking third-party clients is just a symptom of centralization, and App.net just happens to solve this one symptom without addressing the root problem.
Any twitter follower you want to collaborate with would have to make a new Dropbox account, and that account wouldn't have access to any data that she already stored in a different application. So even if you made an app to bridge all these services, it would still be a huge pain.
People could use their existing DB accounts, I'm sure.
Centralization is a problem because it gives a single entity total control. That's why Twitter was able to screw with third-party clients in the first place, and why it mattered that they did it. Take e-mail as a counterexample: Google, for example, couldn't really screw with third-party e-mail clients, and even if they did, people who want to use third-party clients can just switch to a provider that isn't stupid.
App.net retains all of this control. They merely promise not to abuse it. I'd much rather not have to trust anyone's promises. Even if they remain completely honest, their idea of abuse may not match my own.
Uh... not everyone who has a twitter account also has a Dropbox account. But everyone who has an app.net account has a Patter account, and an ADN File Manager account, etc.