Such a global issue with playback is just bizarre and definitely not normal behaviour for any Tablo. It sounds like a networking issue.
I would open up a Support Ticket with Tablo directly. They likely can give you step by step help that you need.
The Tablo plays back smoothly on 2 HTPCs, 2 Roku 3s, iPhone 6, iPad Air, Android tablet and now Fire TV Stick. So the Tablo works if it is across so many devices.
I will admit I had buffering issues (not stuttering issues) when I first got my Fire TV Stick but my old router was only Wireless N 2.4 GHz. I upgraded my router and put the Stick on the 5 GHz band and my issues on the Fire TV Stick went away. The router is great and it was only $50, TL-WDR3600.
I'm sure your issues can be resolved if you work on it - good luck!