This work combines the techniques of inverse modelling with in-situ MRI measurements that were performed on a working graphite anode to investigate fundamental accuracy limitations of various models of Li transport within the insertion material. We augment the Newman model with two different descriptions of Lithium transport internal to the graphite, namely (non)linear diffusion and Cahn-Hilliard dynamics. These models are then calibrated via inverse modelling which identifies the transport properties (diffusivities in the diffusion model, or coefficients in the chemical potential in the Cahn-Hilliard model) leading to the best possible agreement between model predictions and measurement data. We find that a Fickian diffusion model is inadequate because it cannot accurately capture the sharp phase transitions of the graphite. We demonstrate that it is however possible to recover accuracy in a diffusion model by increasing its complexity and allowing the diffusivity to be a general function of concentration. When sufficient flexibility is allowed, the inverse modelling recovers a concentration-dependent diffusivity which is in qualitative agreement with the literature and exhibits significant “dips” at values of the concentration known to correspond to a phase change. The Cahn-Hilliard model is also able to accurately fit the measurement data and it does so without the need to allow for state-dependence of its transport properties, indicating that it is overall a more economical choice for modelling intercalation into graphite. Our findings underscore the usefulness of performing model calibration simultaneously based on concentration and cell voltage measurements which can be efficiently done using methods of multi-objective optimization.