In theory one could build a DropBox client that gets you any file that exists on the DropBox cloud.
You just need to have the correct hashes.
Before DropBox uploads a file, the DropBox server compares the hashes of the file with all the hashes of all the files stored in their cloud. If the hashes match, the file's content is not uploaded, because DropBox assumes that the file on the cloud is the same one you have in your DropBox folder.
From this point on, DropBox thinks that you do own the actual file, but the servers can not actually verify that fact. So when you install DropBox on another computer with a _regular_ client, DropBox will be glad to download that file to your folder directly from their cloud.
Update: Yes, it can be done, and actually someone wrote a software that does exactly that: Sharing files on the Dropbox cloud by only transmitting hashes.
Dropbox reacted and seems to have disabled deduplication a few months ago.