Extreme events and critical transitions occur in a variety of natural and artificial systems. The most crucial task for understanding and explaining extreme events is usually the question of predicting these events. The notion of self-organized criticality is often used to explain these events. However, here the underlying mechanisms of extreme events are the same as for all other events which makes prediction difficult. Recently the novel concept of Dragon Kings was introduced, where extreme events have a distinct mechanism from all other events. Here we introduce a simple network model where nodes self-organize to be either weakly or strongly protected against failure in a manner that captures the trade-off between degradation and reinforcement of nodes inherent in many network systems. If strong nodes cannot fail, any failure is contained to a single, isolated cluster of weak nodes and the model produces power-law distributions of failure sizes. We classify the large, rare events that involve the failure of only a single cluster as “Black Swans.” In contrast, if strong nodes fail once a sufficient fraction of their neighbors fail, then failure can cascade across multiple clusters of weak nodes. If over 99.9% of the nodes fail due to this cluster hopping mechanism, we classify this as a “Dragon King,” which are massive failures caused by mechanisms distinct from smaller failures. We find that once an initial cluster of failing weak nodes is above a critical size, the Dragon King mechanism kicks in, leading to piggybacking system-wide failures. We demonstrate that the size of the initial failed weak cluster predicts the likelihood of a Dragon King event with high accuracy and we develop a simple control strategy that can dramatically reduce Dragon Kings and other large failures.