What I don't like about the example you provide is: what player 1 and player 2 know needs to be common knowledge. For instance if player 1 doesn't know whether player 2 knows whether die 1 is in 1-3, then it may not be common knowledge at all that the sum is in 2-6, even if player 1 and player 2 are given the info you said they're given.

This is what I was confused about in the grandparent comment: do we really need I and J to be common knowledge? It seems so to me. But that seems to be another assumption limiting the applicability of the result.