Thick neutron skins (approximately 0.9 fm) have been verified for the first time in He-6 and He-8 nuclei from a combined analysis of the interaction and the fragmentation cross sections of He-4, He-6, He-8 incident reactions at 800A MeV. A relativistic mean field (RMF) model was applied to He-6 and He-8 nuclei and shown to reproduce the neutron skin thicknesses very well. It is also shown that the RMF model predicts a gross linear dependence of the neutron skin thickness on the difference between the proton and the neutron Fermi energy in a wide range of nuclei. Possible observations of thick neutron skins in other nuclei, in particular in Na isotopes, are also discussed.