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I have 3 Rokus in my house and HBO Go is installed on all 3 of them. Starting with the recent "new" app upgrade I've ran into an issue. If I activate only 2 of them there's no problem but if I activate the 3rd one the first two get logged out. Then if I reactivate the 1st two the 3rd one gets logged out. And I'm not talking simultaneous streams, I'm talking just being logged in to see the watch list, etc. I don't have this issue with any other providers so is Cox limiting how many devices can be used?
I re-read your OP, and I think I have a different idea. The fact that it isn't giving you the typical "Too many users" message, and the fact that it's signing out BOTH units, not just the possible 4th one, makes me think something else is going on. I wonder if your router or network is only able to keep the connection open for 2 devices at once. Can you try accessing the HBO Go app from your phone when NOT on Wifi? I wonder if it will allow the 3rd stream if not initiated on your local network.