The shift to running application software on the server side has obviously simplified development and deployment a lot. Storing the data server-side is a natural thing to do also. But the problem with this setup is that the user has less control. The data is in the hands of the application provider.
I'm working on a music player app for Android as part of a research project. However, the app can only access music stored in the device's central media database. A lot of users have their music stored away in Google Music, Amazon Music, etc. My app doesn't work for them.
What if everyone used Dropbox to handle data syncing and web applications didn't bother about data storage? I don't see any reason why web applications like Google Docs couldn't operate on local files. There may be security concerns since the applications would need to interact more with your file system, but that shouldn't be a show stopper. There wouldn't be a loss of convenience since Dropbox would sync the data for you, but the user would have more control since their data wouldn't be tied to any particular software. They wouldn't even have to depend on Dropbox if a good competitor came along.
Dropbox does need some improvements, especially for mobile devices. Currently (on Android, at least) the official app doesn't actually sync the files to your phone's file system. The files are only available from within the Dropbox app. There is an app to fix that, and it works wonderfully. I can take a picture with my phone and it immediately gets synced to the file system on my workstation. However, it'd be better if we didn't have to rely on a third party for this.
But as issues like that get ironed out, I think it could work great to factor out data syncing into a single service like Dropbox. All your data would be consolidated on your file system instead of being spread out across a bunch of separate applications, and developers (like me) wouldn't have to deal with implementing data syncing.