There has been much speculation about the origin of long-range electrostatic attractions between identical colloidal particles in confined geometries. Recently, we proved that such attractive interactions are not to be found in the well-established Poisson-Boltzmann theory, when the particles are immersed in a 1:1 electrolyte whose average ion concentrations are equal. A subsequent approximate analytical investigation (Europhys. Lett. 1999, 46, 407-413) has suggested that such attractive interactions result from a combination of the effects of confinement, imbalance of the average ionic concentrations, and polarization effects in the confining surface. Consequently, we extend our previous proof to encompass the general case of an electrolyte possessing any number of ionic species, where there is no restriction on their average concentrations. In so doing, we rigorously prove that within the framework of the Poisson-Boltzmann theory the interaction between identical colloidal particles is never attractive, irrespective of whether the particles are isolated or confined. Furthermore, we establish a necessary condition for the existence of attractive interactions, which indicates the possibility that an osmotically driven process is behind the observed attractive interactions.