It has been recently shown that general policies for many clas-sical planning domains can be expressed and learned in termsof a pool of features defined from the domain predicates usinga description logic grammar. At the same time, most descrip-tion logics correspond to a fragment of k-variable countinglogic (Ck ) for k = 2, that has been shown to provide a tightcharacterization of the expressive power of graph neural net-works. In this work, we make use of these results to under-stand the power and limits of using graph neural networks(GNNs) for learning optimal general policies over a numberof tractable planning domains where such policies are knownto exist. For this, we train a simple GNN in a supervised man-ner to approximate the optimal value function V ∗(s) of anumber of sample states s. As predicted by the theory, it is ob-served that general optimal policies are obtained in domainswhere general optimal value functions can be defined withC2 features but not in those requiring more expressive C3 fea-tures. In addition, it is observed that the features learned are inclose correspondence with the features needed to express V ∗in closed form. The theory and the analysis of the domainslet us understand the features that are actually learned as wellas those that cannot be learned in this way, and let us movein a principled manner from a combinatorial optimization ap-proach to learning general policies to a potentially, more ro-bust and scalable approach based on deep learning.