While the experimental H-1 NMR chemical shifts of the 1-adamantyl cation can be computed within reasonably small error bounds, the usual Hartree-Fock and density functional quantum chemical computations, as well as those based on rather elaborate second order Moller-Plesset perturbation theory, fail to reproduce its experimental C-13 NMR chemical shifts satisfactorily. This also is true even if the NMR shielding calculations treat electron correlation adequately by the coupled-cluster singles and doubles model augmented by a perturbative correction for triple excitations (i.e., at the CCSD(T) level) with quadruple-zeta basis sets. We demonstrate that good agreement can be achieved if highly accurate 1-adamantyl cation equilibrium geometries based on parallel computations of CCSD(T) gradients are employed for the NMR shielding computations.