Kerr-cat qubits have been experimentally shown to exhibit a large noise bias, with one decay channel suppressed by several orders of magnitude. In superconducting implementations, increasing the microwave drive on the nonlinear oscillator that hosts the Kerr-cat qubit should, in principle, further enhance this bias. Instead, experiments reveal that above a critical drive amplitude the tunneling time, which is the less dominant decay channel, ceases to increase and even decreases. Here, we show that this breakdown arises from the multimode nature of the circuit implementation. Specifically, additional modes, including the buffer mode used to deliver the stabilizing drive and higher modes of the Josephson junction array, can induce multiphoton resonances that sharply degrade Kerr-cat coherence. We uncover this mechanism by retaining the full circuit nonlinearities and treating the strong drive exactly within a Floquet–Markov framework that accounts for quasidegeneracies of the Kerr-cat spectrum. Our results not only provide an explanation for the sudden reduction of the tunneling time but also demonstrate that the Kerr-cat qubit can be very robust in the presence of a carefully engineered electromagnetic environment. Beyond the Kerr-cat qubit, the tools developed here open the door to study a broad range of drive-induced phenomena in dissipative superconducting circuits, in both resonant and off-resonant driving regimes. Examples include inelastic scattering in circuits with arbitrary anharmonicity and drive strength, as well as subharmonically driven devices such as parametric amplifiers and couplers, and other protected qubits whose coherence is governed by quasidegeneracies.