An inelastic neutron scattering study of Cu2Te2O5X2 (X=Cl, Br) shows strong dispersive modes with large energy gaps persisting far above T-N, notably in Cu2Te2O5Br2. The anomalous features: a coexisting unusually weak Goldstone-like mode observed in Cu2Te2O5Cl2 and the size of the energy gaps cannot be explained by existing theories, such as our mean-field or random-phase approximation. We argue that our findings represent a new general type of behavior due to intercluster quantum fluctuations and call for development of a new theoretical approach.