The mean-field ENMO (Electronic and Nuclear Molecular Orbitals) wave function is written as a product of one-particle functions that pertain to different types of particles, antisymmetrized over the fermionic orbitals, thus allowing the simultaneous treatment of electrons and nuclei within a self-consistent framework. We implemented the ENMO-SCF (self-consistent field) as well as correlated versions of ENMO analogous to the MBPT (many-body perturbation theory) and CI (configuration interaction) models of Born-Oppenheimer quantum chemistry. The ENMO methods are attractive by virtue of their similarity to traditional electronic structure theory models, and several researchers have recently implemented such approaches. Here we present the first fundamental study of the convergence of ENMO methods with respect to basis set and correlation treatment for simple systems. It is found that the elimination of the translational invariance of the full molecular Hamiltonian is pivotal for obtaining a physically meaningful spectrum. For example, the ENMO-Full CI spectrum for hydrogen atom contains a manifold of spurious states which cannot be removed by simple subtraction of centre-of-mass kinetic energy due to the general non-factorizability (non-separability) of the ENMO ansatz into internal and centre-of-mass parts. An alternative, hybrid approach that treats only some light nuclei in a molecule quantum-mechanically avoids the spurious states problem by breaking the translational invariance of the Hamiltonian. However, both simple analytical estimations and numerical examples for some diatomic hydrides show that vibrational energy levels are not accurately predicted in this approach; this suggests difficulty in using such methods to predict tunnelling splittings. Finally, slow convergence with respect to basis set and correlation method are observed and related to the failure of ENMO methods to describe interparticle cusps (including the electron/nuclear cusp) efficiently due to the lack of explicit interparticle coordinates. We elevate these previously unrecognized or underappreciated difficulties as being of prime importance to future progress in ENMO methods.