Internet of Things (IoT) is a networking paradigm that interconnects physical systems to the cyber world, to provide automation and intelligence via interdependent links between the two domains. Such interdependence renders IoT systems vulnerable to random failures, e.g., broken communication links or crashed cyber instances, because a single incident in one domain can develop into a <italic>cascade-of-failures</italic> across domains, which dissolves the network structure, and has devastating consequences. To answer how robust an IoT system is, this paper studies its <italic>resilience</italic> by examining the impact of edge- and jointly-induced cascades, that is, a sequence of failures caused by randomly broken physical links (and simultaneous failing cyber nodes). Resilience of an IoT system is quantified by two new metrics, the <italic>critical edge disconnecting probability</italic> <inline-formula> <tex-math notation="LaTeX">$\phi _{cr}$ </tex-math></inline-formula>, i.e., the maximum intensity of random failures the system can withstand, and the cascade length <inline-formula> <tex-math notation="LaTeX">$\tau _{cf}$ </tex-math></inline-formula>, i.e., the lifetime of a cascade. For IoT systems with Poisson degree distributions, we derive exact solutions for the critical disconnecting probability <inline-formula> <tex-math notation="LaTeX">$\phi _{cr}$ </tex-math></inline-formula>, above which an edge-induced cascade will completely fragment the network. We also find that the critical condition <inline-formula> <tex-math notation="LaTeX">$\phi _{cr}$ </tex-math></inline-formula> marks a dichotomy of the expected cascade length <inline-formula> <tex-math notation="LaTeX">$\mathbb {E}(\tau _{cf})$ </tex-math></inline-formula>: for the super-critical (<inline-formula> <tex-math notation="LaTeX">$\phi > \phi _{cr}$ </tex-math></inline-formula>) scenario, we obtain <inline-formula> <tex-math notation="LaTeX">$\mathbb {E}(\tau _{cf}) \sim \exp (1-\phi)$ </tex-math></inline-formula> through analysis, while for the subcritical scenario, we observe <inline-formula> <tex-math notation="LaTeX">$\mathbb {E}(\tau _{cf}) \sim \exp (1/1-\phi)$ </tex-math></inline-formula> through simulations. With these results, the final outcome of a cascade can be anticipated upon the initial failures, while the reaction window of time-sensitive countermeasures can be obtained before a cascade fully unfolds.