Without Lorentz invariance, spontaneous global symmetry breaking can lead to multicritical Nambu-Goldstone modes with a higher-order low-energy dispersion ω∼kn (n=2,3,…), whose naturalness is protected by polynomial shift symmetries. Here, we investigate the role of infrared divergences and the nonrelativistic generalization of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem. We find novel cascading phenomena with large hierarchies between the scales at which the value of n changes, leading to an evasion of the “no-go” consequences of the relativistic CHMW theorem.