We study the role of informational complexity on the higher-order rationality of agents in Bayesian games of incomplete information. In a class of dominance solvable Bayesian chain games, we find evidence that informational environments that rely on more private information, and are therefore more complex in terms of contingent thinking requirements, are associated with lower orders of rationality. The results have implications for the optimal design of information structures, by highlighting a potential source of discrepancy between observed behavior and theoretical predictions. (joint with Brian Rogers)