We calculate the expected fluxes of cosmic ray nuclei with charge 5 </= Z </= 28 at various depths in the earth's atmosphere, taking into account the initial charge distribution, ionization loss, and various modes of fragmentation. The flux of surviving heavy nuclei is too low by a factor approximately 10(-10) to account for the ultra-high-energy Centauro events. We describe an experiment to search for highly ionizing particles that may or may not be nuclei.