Exchange collisions of electrons with Na and Hg atoms and O2 and NO molecules have been studied by means of polarized electrons for energies between 4 and 15 eV and scattering angles ranging from 0° to 110°. While significant exchange effects have been observed for collisions of electrons with Na and Hg atoms, in agreement with theoretical predictions, differential spin-exchange cross sections for elastic collisions from the open-shell molecules O2 and NO are much smaller. This was not anticipated before. A satisfactory explanation of this different behavior is still missing.