The tribute cap is 15 or higher: I got a 15-kingdom tribute last week. With 29 kingdoms, at 26% chance each, the odds of a 16 or higher are 1 in 1257. Even if I were collecting 12 tributes a day, that is three months of collections before I should expect to have seen a 16+.
Now, once we start seeing 9-starred kingdoms, the math changes dramatically. But for now a cap of 15 (or whatever it is) is highly unlikely to impact play in a significant way. You would lose a couple of gems once every thousand plus tribute collections.
Edit: this, of course, is assuming a tribute cap even exists. Has a dev said anything of the sort?
Edit again: there was a question upthread about a mobile app (a la Mass Effect) that would let console players log in and collect tribute. A worthy suggestion, I think.