I just disconnected my WAN to see what would happen... Initially, I could connect with my Roku and even play live TV, until I reset it, then lost the ability to connect at all. On my browser (pointed to my.tablotv.com) I couldn't connect, though when I pointed my browser to the numeric IP of the Tablo, I got the Nuvyyo server message only. My android Asus Nexus Player could not connect at all. Didn't try my Android cell phone, but I suspect the results would not be different.
So my test shows that the Tablo is still reachable on the LAN, it just doesn't function. That implies that our LAN peripherals don't communicate directly with Tablo, instead, requiring that we access it via the Tablo servers in order to work.
It's odd that we totally lose the ability to watch live or recorded TV if the link to the server breaks for any reason. Might be nice if they could fix that.
Anyway, back to your problem, it's possible that you're one of the few people who experiences frequent WAN IP changes (I don't think mine has ever changed, or maybe rarely even tho I have a dynamic IP) so your situation may not be all that common, and it possibly points to a bug in the Tablo's code, where maybe it needs to "phone home" more frequently so the servers know where it is.