Traditionally, studies aimed at inferring the distribution of birth periods of neutron stars are based on radio surveys. Here we propose an independent method to constrain the pulsar spin periods at birth based on their X-ray luminosities. In particular, the observed luminosity distribution of supernovae poses a constraint on the initial rotational energy of the embedded pulsars, via the L_X-dot{E}_{rot} correlation found for radio pulsars, and under the assumption that this relation continues to hold beyond the observed range. We have extracted X-ray luminosities (or limits) for a large sample of historical SNe observed with Chandra, XMM and Swift, that have been firmly classified as core-collapse supernovae. We have then compared these observational limits with the results of Monte Carlo simulations of the pulsar X-ray luminosity distribution, for a range of values of the birth parameters. We find that a pulsar population dominated by millisecond periods at birth is ruled out by the data.