Are you sure you are using CC 2015.4? There was definitely a tethering problem with earlier versions.
Try all these solutions. https://helpx.adobe.com/lightroom/kb/troubleshoot-tethered-capture-lightroom-4.html
For me, when I updated to Lr CC 2015.4 tether on my Canon 5D Mk3 quit working. Resetting preferences was the solution that worked. Adobe REALLY needs to make error checking during creation/changing of the preferences more robust. Its amazing what an invalid preference file will mess up. This preference problem has been going on since Lightroom was first created. This is not only a Lightroom problem. Some of their other programs like Photoshop Elements are affected by corrupt preference files also.
Yes, I'm sure of the version. I looked it up in the "about" screen of Lightroom CC as I was typing the question. I just now when to the Lightroom control panel and selected the "check for updates" option and it said all my apps were up-to-date (see screenshots below).
Sent from my iPad
Thank you thank you! Clearing the plist (presets) files and connecting/starting in the correct order fixed the problem.