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Network structures in a wide array of systems such as social networks, transportation, power and water distribution infrastructures, and biological and ecological systems can exhibit critical thresholds or tipping points beyond which there are disproportionate losses in the system functionality. There is growing concern over tipping points and failure tolerance of such systems as tipping points can lead to an abrupt loss of intended functionality and possibly non-recoverable states. While attack tolerance of networked systems has been intensively studied for the disruptions originating from a single point of failure, there have been instances where real-world systems are subject to simultaneous or sudden onset of concurrent disruption at multiple locations. Using open-source data from the United States Airspace Airport network and Indian Railways Network, and random networks as prototype class of systems, we study their responses to synthetic attack strategies of varying sizes. For both types of networks, we observe the presence of warning regions, which serve as a precursor to the tipping point. Further, we observe the statistically significant relationships between network robustness and size of simultaneous distribution, which generalizes to the networks with different topological attributes for random failures and targeted attacks. We show that our approach can determine the entire robustness characteristics of networks of disparate architecture subject to disruptions of varying sizes. Our approach can serve as a paradigm to understand the tipping point in real-world systems, and the principle can be extended to other disciplines to address critical issues of risk management and resilience.