Kindle newbie here (Sony orphan making the switch). Decided to see what would happen if I tried to access Dropbox and ... it worked !?!
I have a new Kindle Voyage with latest firmware 5.6.1 accessing the mobile Dropbox site.
The browser's a little flaky and I often have to press the link a few times before it takes, but I have successfully downloaded several files from Dropbox to Kindle yesterday and today. Hope it's not a fluke