Dropbox has a lovely feature in which they retain multiple versions of your files in case you want to revert some recent changes you have made. When you log into their website they let you pick from approximately all of last month's versions of a given file. However, if you look at URL of the version previews they offer, there is something to notice; they look roughly like:

https://dl-web.dropbox.com/get/path/to/your/file.txt?w=abcdef&sjid=123456

The w=abcdef is the ID of your file, and the sjid=123456 is the version of that file. Even though the web interface only shows a limited number of versions, since the version numbers appear to start counting at 1 you can simply contruct your own URLs to grab versions that they would not normally allow.

If you are especially careful and copy your browser cookies you can even automate this process to grab all versions of a given file!