Typical strategies for reducing the computational cost of contact mechanics models use low-rank approximations. The underlying hypothesis is the existence of a low-dimensional subspace for the displacement field and a { non-negative low-dimensional subcone for the contact pressure}. However, given the local nature of contact, it seems natural to wonder whether low-rank approximations are a good fit for contact mechanics or not. In this paper, we investigate some of their limitations and provide numerical evidence showing that contact pressure is linearly inseparable in many practical cases. To this end, we consider various mechanical problems involving non-adhesive frictionless contacts and analyse the performance of the low-rank models in terms of three different criteria, namely compactness, generalization and specificity.