It's been a little while that I made my first tests with the DropBox repository; at that time I was using Moodle 2.0 and hadn't any problems with it.
Today, using Moodle 2.0.2, I tried to add a file resource through my DropBox, and after clicking on the log in button, I got the following message:
Wrong authorization URL
For the best user experience, you should be sending users to https://www.dropbox.com/0/oauth/authorize, as per the documentation.
I don't know, but it seems that somewhere in the Moodle code the URL needs to be updated. Right now the file picker opens a window with the following address:
https://api.dropbox.com/0/oauth/
whereas it should be the next one:
https://www.dropbox.com/0/oauth/
To be able to access my DropBox files, I had to copy the URL, open a new browser tab, paste the address, change the "api" part to "www" and press Enter. A message then appeared saying something about the way the window had been opened, but that everything was OK and that I should be able to use the file picker without any problems. After returning to my Moodle tab and selecting the DropBox repository, everything went fine.
Cheers!