From the perspective of many body physics, the transmon qubit architectures currently developed for quantum computing are systems of coupled nonlinear quantum resonators. A certain amount of intentional frequency detuning (`disorder') is crucially required to protect individual qubit states against the destabilizing effects of nonlinear resonator coupling. Here we investigate the stability of this variant of a many-body localized phase for system parameters relevant to current quantum processors developed by the IBM, Delft, and Google, considering cases of both natural or engineered disorder.  Applying three independent diagnostics of localization theory --- a Kullback-Leibler analysis of spectral statistics, statistics of many-body wave functions (inverse participation ratios), and a Walsh transform of the many-body spectrum --- we find that some of these computing platforms are dangerously close to a phase of uncontrollable chaotic fluctuations.