In many leading architectures for quantum computing, it remains to be understood if we can equate single-qubit coherence times measured in isolation, to that of coherence times measured in multi-qubit devices. On a multi-qubit superconducting circuit platform we show an increase in the dephasing rate of a control qubit due to the spontaneous relaxation of spectator qubits coupled to the control qubit. We attribute this increased dephasing to random in time Z-phase kicks on the control qubit due to the interplay between spectator relaxation and the control-spectator ZZ-interaction. We measure the magnitude of this extra dephasing using Ramsey decay experiments, show how it can be corrected via dynamical decoupling pulse sequences, and demonstrate that randomized benchmarking is insensitive to the effect. Our experimental results are supported by a robust theoretical model that captures an arbitrary number of spectator qubits, and gives a simple, intuitive picture for the mechanism behind the enhanced dephasing.