I have/had this issue on my Fire TV and Fire TV Stick and all 3 of my Nexus Players. At first the stuttering issue was limited, but then all of my devices that were working fine previously started with the stuttering as well.
I fixed my Android TVs by uninstalling the 1.6.2.0 Tablo Video Player app and sideloading 1.5.0.26 version from my phone. I haven't done anything to try to fix the issue on my Fire TV boxes yet because I don't care much about those anymore.
None of my devices are rooted.
Nexus Player OS (All): Android 5.1
Fire TV OS: 51.1.5.0_user_515030720
Fire TV Stick OS: 54.1.1.0_user_110041020
Tablo Firmware: 2.1.28
Tablo app version: 1.0.15-477/10090
It is really hard to judge how many people are having this issue as the suggestion to roll back to a previous version of the video player app seems to have fixed the issue for so many.