Measurements of thermal diffusivity in several insulators have been shown to reach a Planckian bound on thermal transport that can be thought of as the limit of validity of semiclassical phonon scattering. Beyond this regime, the heat transport must be understood in terms of incoherent motion of the atoms under strongly anharmonic interactions. In this work, we propose a model for heat transport in a strongly anharmonic system where the thermal diffusivity can be lower than the Planckian thermal diffusivity bound. Similar to the materials that exhibit thermal diffusivity close to this bound, our scenario involves complex unit cells with incoherent intra-cell dynamics. We derive a general formalism to compute thermal conductivity in such cases with anharmonic intra-cell dynamics coupled to nearly harmonic inter-cell coupling. Through direct numerical simulation of the nonlinear unit-cell motion, we explicitly show that our model allows sub-Planckian thermal diffusivity. We find that the propagator of the acoustic phonons becomes incoherent throughout most of the Brillouin zone in this limit. We expect these features to apply to more realistic models of complex insulators showing sub-Planckian thermal diffusivity, suggesting a multispecies generalization of the thermal diffusivity bound that is similar to the viscosity bound in fluids.