The purpose of this paper is to investigate to what extent cooperativity, that is, the absence of negative interactions, in Boolean networks with synchronous updating, imposes limits on chaos-like properties that are possible in such systems. Our focus is on notions of sensitive dependence on initial conditions, or a combination of sensitive dependence and large basins of attraction of exponentially long attractors, both of which are well-recognized hallmarks of chaotic dynamics in the Boolean context.We prove that a strong notion of sensitive dependence on initial conditions that formalizes decoherence along the attractor is precluded by cooperativity. Weaker notions of sensitive dependence that formalize decoherence at some time during the trajectory and sensitive dependence of the basin of attraction on initial conditions, respectively, are shown to be consistent with cooperativity, but if each regulatory function is binary AND or binary OR, in N-dimensional networks they impose an upper bound of ≈ 3 N on the lengths of attractors that can be reached from a fraction p≈1 of initial conditions. The upper bound is shown to be optimal. These results indicate that the transfer of analogous results for differential equations models crucially depends on the precise conceptualization of chaos in the Boolean context.MSC: 34C12, 39A33, 94C10.