Nanoscale systems of metal atoms antiferromagnetically exchange coupled to several magnetic impurities are shown to exhibit an unconventional reentrant competition between Kondo screening and indirect magnetic exchange interaction. Depending on the atomic positions of the magnetic moments, the total ground-state spin deviates from predictions of standard Ruderman-Kittel-Kasuya-Yosida perturbation theory. The effect shows up on an energy scale larger than the level width induced by the coupling to the environment and is experimentally accessible by studying magnetic field dependencies.