This study aims to identify the minimum requirements for an accurate model of the diffusion MR signal in white matter of the brain. We construct a hierarchy of two-compartment models of white matter from combinations of simple models for the intra and extracellular spaces. We devise a new diffusion MRI protocol that provides measurements with a wide range of parameters for diffusion sensitization both parallel and perpendicular to white matter fibres. We use the protocol to acquire data from a fixed rat brain, which allows us to fit, study and compare the different models. The results show that models which incorporate pore size describe the measurements most accurately. The best fit comes from combining a full diffusion tensor (DT) model of the extra-cellular space with a cylindrical intra-cellular component.