How to get the managed_file progress bar working in a custom theme?
Found, I think: My theme was based on Bootstrap 2, and I was getting bit by a problem of Drupal and Bootstrap disagreeing on what sort of thing the "Upload" button should be:

https://drupal.org/node/2155419

Since I'm still on Bootstrap 2 (I think) I wasn't having the problem with the input fields being out of order, but I did need to deal with the behavior assignments.  In addition to the fix outlined in the cited page, I also needed to run the assignments on page load / document ready to get things properly assigned.