Deep neural networks are shown to be susceptible to both adversarial attacks and backdoor attacks. Although many defenses against an individual type of the above attacks have been proposed, the interactions between the vulnerabilities of a network to both types of attacks have not been carefully investigated yet. In this paper, we conduct experiments to study whether adversarial robustness and backdoor robustness can affect each other and find a trade-off—by increasing the robustness of a network to adversarial examples, the network becomes more vulnerable to backdoor attacks. We then investigate the cause and show how such a trade-off can be exploited for either good or bad purposes. Our findings suggest that future research on defense should take both adversarial and backdoor attacks into account when designing algorithms or robustness measures to avoid pitfalls and a false sense of security.