In this paper we investigate how observable cognitive skills influence the development of strategic sophistication. To answer this question, we study experimentally how psychometric measures of theory-of-mind and cognitive ability (or 'fluid intelligence') work together with age to determine the strategic ability and level-k behavior of children in a variety of incentivized strategic interactions. We find that better theory-of-mind and cognitive ability predict strategic sophistication in competitive games. Furthermore, age and cognitive ability act in tandem as complements, while age and theory-of-mind operate independently. Older children respond to information about the cognitive ability of their opponent, which provides support for the emergence of a sophisticated strategic theory-of-mind. Finally, theory-of-mind and age strongly predict whether children respond to intentions in a gift-exchange game, while cognitive ability has no influence, suggesting that different psychometric measures of cognitive skill correspond to different cognitive processes in strategic situations that involve the understanding of intentions.