Neural networks are known to be susceptible to adversarial examples. Different techniques have been proposed in the literature to address the problem, ranging from adversarial training with robustness guarantees to post-training and run-time certification of local robustness using either inexpensive but incomplete verification or sound, complete, but expensive constraint solving. We advocate for the use of a run- time cascade of over-approximate, under-approximate, and exact local robustness checkers. The exact check in the cascade ensures that no unnecessary alarms are raised, an important requirement for autonomous systems where resorting to fail-safe mechanisms is highly undesirable. Though exact checks are expensive, via two case studies, we demonstrate that the exact check in a cascade is rarely invoked in practice.