The relationship between antiferromagnetic spin fluctuations and superconductivity has become a central topic of research in studies of superconductivity in the iron pnictides. We present unambiguous evidence of the absence of magnetic fluctuations in the nonsuperconducting collapsed tetragonal phase of CaFe2As2 via inelastic neutron scattering time-of-flight data, which is consistent with the view that spin fluctuations are a necessary ingredient for unconventional superconductivity in the iron pnictides. We demonstrate that the collapsed tetragonal phase of CaFe2As2 is nonmagnetic, and discuss this result in light of recent reports of high-temperature superconductivity in the collapsed tetragonal phase of closely related compounds.