Description: We combine neutron scattering (INS) and NMR/NQR nuclear spin lattice relaxation rate data to deduce the existence of a new contribution to the magnetic response (dynamic susceptibility) in cuprate superconductors. This contribution, which has yet to be observed with INS, is shown to embody the magnetic pseudogap effects. As such, it explains the long-standing puzzle of pseudogap effects missing from cuprate INS data, dominated by stripe fluctuations, for the dynamic susceptibility at low energies. For La_(1.86)Sr_(0.14)CuO_(4) and YBa_(2)Ba_(3)O_(6.5), the new term is the chief contributor to nuclear spin lattice relaxation at T >> T_(c). *** Journal reference: "Phys. Rev. B84, 024530 (2011)", http://prd.aps.org/ ***