A natural way for cooperative tasking in multi-agent systems is through a top-down design by decomposing a global task into subtasks for each individual agent such that the accomplishments of these subtasks will guarantee the achievement of the global task. In our previous works [Karimadini, M., and Lin, H. (2011c), 'Guaranteed Global Performance Through Local Coordinations', Automatica, 47, 890-898; Karimadini, M., and Lin, H. (2011a), 'Cooperative Tasking for Deterministic Specification Automata', submitted for publication, online available at: http://arxiv.org/abs/1101.2002], we presented necessary and sufficient conditions on the decomposability of a global task automaton between cooperative agents. As a follow-up work, this article deals with the robustness issues of the proposed top-down design approach with respect to event failures in the multi-agent systems. The main concern under event failure is whether a previously decomposable task can still be achieved collectively by the agents, and if not, we would like to investigate that under what conditions the global task could be robustly accomplished. This is actually the fault-tolerance issue of the top-down design, and the results provide designers with hints on which events are fragile with respect to failures, and whether redundancies are needed. The main objective of this article is to identify necessary and sufficient conditions on failed events under which a decomposable global task can still be achieved successfully. For such a purpose, a notion called passivity is introduced to characterise the type of event failures. The passivity is found to reflect the redundancy of communication links over shared events, based on which necessary and sufficient conditions for the reliability of cooperative tasking under event failures are derived, followed by illustrative examples and remarks for the derived conditions.