Solutions of highly charged polyelectrolyte chains are described by a model that introduces ion condensation as a random charge along the polymer. The degree of condensation is obtained by solving the Poisson-Boltzmann equation with cylindrical geometry. Short range electrostatic attractions between the monomers via the condensed counterions of high enough valency lead to reversible chain precipitation. The range of polymer concentration over which salt-free solutions are unstable is determined, as well as the miscibility of the chains when salt is added. Redissolution at high salt concentration is due to a screening of the short range electrostatic attractions. Precipitation of chains in mixtures of movalent and multivalent salts is also studied. We find the range of salt concentration where chains precipitate. The model explains the experimental results on the precipitation of sodium and lanthanum polystyrene sulfonate solutions in presence of multivalent salts [LaCl3 and Th(NO3)(4)].