Present economic theories make a common-knowledge assumption that implies that the first or the second-order beliefs determine all higher order beliefs. We analyze the role of such closing assumptions at finite orders by instead allowing higher orders to vary arbitrarily. Assuming that the space of underlying uncertainty is sufficiently rich, we show that the resulting set of possible outcomes, under an arbitrary fixed equilibrium, must include all outcomes that survive iterated elimination of strategies that are never a strict best reply. For many games, this implies that, unless the game is dominance solvable, every equilibrium will be highly sensitive to higher-order beliefs, and thus economic theories based on such equilibria may be misleading. Moreover, every equilibrium is discontinuous at each type for which two or more actions survive our elimination process. Keywords: higher-order uncertainty, rationalizability, incomplete information, equilibrium, robustness. JEL Classifications: C72, C73.

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