EDIT: Given the replies below, I should be clear that I'm not interested in comparing to LastPass, I'm comparing to Bitwarden. LastPass had an obviously bad security model that failed to encrypt everything, but Bitwarden does not have that flaw.
Dropbox is not added security in this setup, it is a natural factor if what is being transferred [the keepass file(s)] is sufficiently secure in itself.
Another reply indicates that the main thing is that you don't have to trust the cloud service to do the encryption and zero-knowledge stuff right.
That plus security through obscurity: no one is presuming you're going to come out of a Dropbox hack with millions of password vaults. Even finding them would be... nightmarish. (Though I suppose you could somehow hack a Dropbox file index database?) The value of a target like LastPass is absolutely insanely high: it's a concentrated honeypot of encrypted vaults.
Plus, the Android app makes using a Dropbox synced folder location fairly trivial, so that works pretty well. And you can set your own number of password rotations, which, while annoying when it takes my phone 5-10 seconds to unlock, realllllllly helps ensure no one else is going to crack this vault if they ever got it.
But what you said is also an additional benefit.
That said, the same risk applies to any client you use. Someone could have compromised the latest update of KeePassX as readily as they can compromise LastPass's client. If you don't have automatic updates then that's helpful, but I'm not sure it's producing enough security to be worth the extra hassle.
Assuming that I trust Bitwarden not to lie about their security model, what do I gain by piecing together multiple tools to accomplish the same thing?
In case of keepass and independent sync(doesn't have to be Dropbox), software that sees master password doesn't need access to the internet. Can be even airgapped if you are extra paranoid.
So to sum it up: keepass + sync is better, because there's no single party that is even able to screw up you to the point of leaking your passwords. "Impossible to fail" is better than "they are doing their best, pinkie promise".
Also - why pay recurring fee for yet another cloud storage, when I just need plain encryption software.
The part where they said they do not store either the key or password on Dropbox.
