Can I download a file in Chrome into Dropbox so that it gets uploaded to Dropbox? Is there a way to download a file in Chrome (on my Nexus 7) and then have it uploaded to a folder in Dropbox?  On a desktop PC, I'd just save it into the Dropbox folder and the app would upload it.

Either that, or given the URL to the file somehow get it into my Dropbox?
If you have downloaded a file and want to upload it to dropbox then you can do it inside "Dropbox" app by using "Upload here" option in menu.
or you can use any file manager which provides cloud facility such as "ES File Explorer".
https://play.google.com/store/apps/details?id=com.estrongs.android.pop

or if you do not want to download it locally then there are some websites which directly upload files from url like "http://ctrlq.org/save/"