I think only Sprint has a truly unlimited data plan; Verizon, T-Mobile, even AT&T may claim that they do when in actuality they don't. Their definition of "unlimited" is "unlimited without the amount of data your plan allows". So it could be throttling, could be a port denial.

There was one time that I had to use my cell phone's hot spot to be able to go online (I had accidentally "turned off" that function of the router, which also affected the two iphones being able to go online). Verizon called to inform that they noticed an increased data-usage.

I would try checking to see if the phone is denying the port the game uses, even if manually doing so. If that fails, I would consider that, maybe, your provider is doing some throttling.