We present experimental intrinsic loss power (ILP) values, measured at an excitation frequency of 1 MHz and at relatively low field amplitudes of 3.4 to 9.9 kA/m, as a function of the mean core diameter, for selected magnetic nanoparticle (MNP). The mean core sizes ranged from ca. 8 nm to 31 nm. Transmission electron microscopy indicated that those with smaller core sizes (less than ca. 22 nm) were single-core MNPs, while those with larger core sizes (ca. 29 nm to 31 nm) were multi-core MNPs. The ILP data showed a peak at ca. 20 nm. We show here that this behaviour correlates well with the predicted ILP values obtained using either a non-interacting Debye model, or via dynamic Monte-Carlo simulations, the latter including core-core magnetic interactions for the multi-core particles. This alignment of the models is a consequence of the low field amplitudes used. We also present interesting results showing that the core-core interactions affect the ILP value differently depending on the mean core size.