Spin ice may be considered to be a model system for the investigation of pinch point scattering. We present very-high-resolution numerical simulations and an analytical theory of the pinch point profiles of the near-neighbor and dipolar spin ice models and find these to be in excellent agreement with each other and with existing theory. Most importantly, the pinch points of the dipolar spin ice model are infinitely sharp, as a result of unscreened dipolar fields. These results are compared to polarized neutron scattering measurements of the pinch point profiles in Ho2Ti2O7, considered to be an accurate realization of dipolar spin ice. In contrast to the numerical and analytical results, the experimental pinch point profiles are shown to be broadened in a manner that is quantitatively consistent with fully screened dipolar fields. This striking paradox is not easily resolved: Possible resolutions implicate quantum fluctuations or fundamental corrections to the theory of simulation or polarized neutron scattering. We further discuss our results in the context of spin ice's role as a model Coulomb fluid.