We report on the room temperature thermal conductivity of AlAs-GaAs superlattices (SLs), in which we systematically vary the period thickness and total thickness between 2 - 24 nm and 20.1 - 2,160 nm, respectively. The thermal conductivity increases with the SL thickness and plateaus at a thickness around 200 nm, showing a clear transition from a quasi-ballistic to a diffusive phonon transport regime. These results demonstrate the existence of classical size effects in SLs, even at the highest interface density samples. We use harmonic Atomistic Green’s function calculations to capture incoherence in phonon transport by averaging the calculated transmission over several purely coherent simulations of independent SL with different random mixing at the AlAs-GaAs interfaces. These simulations demonstrate the significant contribution of incoherent phonon transport through the decrease in the transmission and conductance in the SLs as the number of interfaces increases. In spite of this conductance decrease, our simulations show a quasilinear increase in thermal conductivity with the superlattice thickness. This suggests that the observation of a quasilinear increase in thermal conductivity can have important contributions from incoherent phonon transport. Furthermore, this seemingly linear slope in thermal conductivity vs.~SL thickness data may actually be non-linear when extended to a larger number of periods, which is a signature of incoherent effects. Indeed, this trend for superlattices with interatomic mixing at the interfaces could easily be interpreted as linear when the number of periods is small. Our results reveal that the change in thermal conductivity with period thickness is dominated by incoherent (particlelike) phonons, whose properties are not dictated by changes in the AlAs or GaAs phonon dispersion relations. This work demonstrates the importance of studying both period and sample thickness dependencies of thermal conductivity to understand the relative contributions of coherent and incoherent phonon transport in the thermal conductivity in SLs.