We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T +V, where T is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational method for maximizing the entropy subject to the average energy and normalization constraints. The analytical results show i) that the non-extensive thermodynamics formalism should be called into question to explain experimental results described by extended exponential distributions exhibiting long tails, i.e. q-exponentials with q >1, and ii) that in the thermodynamic limit the theory is only consistent in the range 0 q 1 where the distribution has finite support, thus implying that configurations with, e.g., energy above some limit have zero probability, which is at variance with the physics of systems in contact with a heat reservoir. We also discuss the (q-dependent) thermodynamic temperature and the generalized specific heat.