A superconductor of paired protons is thought to form in the core of neutron stars soon after their birth. Minimum energy conditions suggest that magnetic flux is expelled from the superconducting region due to the Meissner effect, such that the neutron star core retains or is largely devoid of magnetic fields for some nuclear equation of state and proton pairing models. We show via neutron star cooling simulations that the superconducting region expands faster than flux is expected to be expelled because cooling timescales are much shorter than timescales of magnetic field diffusion. Thus magnetic fields remain in the bulk of the neutron star core for at least 106 − 107yr. We estimate the size of flux free regions at 107yr to be ≲ 100m for a magnetic field of 1011G and possibly smaller for stronger field strengths.