Android: The default Dropbox app on Android forces you to use only one account at a time. If you have multiple Dropbox accounts and need access, B1 File Manager is a good solution — although it doesn't offer local sync.
By itself, B1 File Manager is a capable app to browse through your Android's files. You can already use multiple Dropbox accounts on a PC. B1 File Manager brings that to Android with the "Add Network" option, where one of the networks is Dropbox. Tap it and you are prompted to sign into a Dropbox account. Once that's set up, you can tap it again to sign into another account.
Since this is all part of a file manager app, it's also much easier to transfer or delete files on any Dropbox accounts you set up.