The present paper is an attempt at a critical analysis of von Neumann's argument, concerning the impossibility of a more complete specification of quantum states, by means of the introduction of supplementary (hidden) variables. The "impossibility proof" of von Neumann (1932) has been considered as a negative and definite answer to the question of hidden variables. However, the validity of the theorem of von Neumann was challenged, especially in the fifties by L. de Broglie, his collaborators, and many other physicists of the realist school. The debate concerning the possibility of a local and causal description of quantum phenomena, received a new impetus after the formulation of the famous inequalities of J. S. Bell (1964). However, the question of the hidden variables remains always open. The same holds for the validity of von Neumann's theorem.