Just upgraded to WP 4.7. Same problem.
I am having the same problem. I can confirm that in my case, it is a brand new install on WP 4.6.1, and full permissions are set for the uploads/buddydrive directory. I am able to load .pdf files and .jpg files, but not Word Docs.
Thanks for looking CrouchingBruin, I actually found that the problem was that I had the vids in iframes. If I drop them in using native WP functionality, the menu works fine.
I also should note that the website is cassandracurtiscounseling.com
I have a similar problem. Looks like the old peekaboo bug on my site, where the dropdown ends up behind a <div> that appears below it on the page. I’m searching for a hack for it.