We show that the PeV neutrinos detected by IceCube put unique constraints on "secret" interactions of neutrinos with the cosmic neutrino background (C nu B). The coupling must be < 0.03 for the mediating boson mass m(X) less than or similar to 2 MeV, g/m(X) < 5 GeV-1 for m(X) greater than or similar to 20 MeV, and g/m(X) < 0.07 Ge V-1 in between. We also investigate the possibility that neutrino cascades degrade high-energy neutrinos to PeV energies by upgrading C nu B where the energy flux of PeV neutrinos can coincide with the Waxman-Bahcall bound or the cosmogenic neutrino flux for protons, thanks to energy conservation. However, a large coupling is required, which is disfavored by laboratory decay constraints. The suppression of PeV-EeV neutrinos is a testable prediction for the Askaryan Radio Array.