After an introduction to the controversial problem of Low Energy Nuclear Reactions (LENR) catalyzed by neutrons on metallic hydride surfaces we present the results of an experiment, made in collaboration with ENEA Labs in Frascati, to search neutrons from plasma discharges in electrolytic cells. The negative outcome of our experiment goes in the direction of ruling out those theoretical models expecting LENR to occur in condensed matter systems under specific conditions. Our criticism on the theoretical foundations of such models will also be presented.