This is a rather obscure problem (I've seen it happen only once in the past 7+ years Chrome has been on the market), but I figured I'd document the solution in case anyone else came across the same problem.
So this user could download documents. A small animation would point downward to indicate a download was in progress. The downloaded document would appear in the actual Downloads folder. But no download status bar would appear on Chrome itself.
So I tried doing a download in an Incognito Window, and that worked fine. So I knew it wasn't the Chrome installation itself that was the problem. It was a user profile thing. So I cleared out the cache for that user... and then it worked fine.
In case you don't know how to clear out the cache in Chrome...
Check only Cached images and files.
Then click Clear browsing data.
Your download status bar should be working again.