Most astronomers now accept that stars more massive than about 9 M_⊙ explode as supernovae and leave stellar remnants, either neutron stars or black holes. However, less than half of the SNRs within 5 kpc have identified central sources. Here, we discuss a systematic effort to search for compact central sources in the remaining 23 SNRs of this distance-limited sample. As the first part of this survey, we are able to state with some confidence that there are no associated central sources down to a level of one tenth of that of the Cas A central source, L_X ≲ 10³¹ ergs s⁻¹, in four SNRs (G093.3+6.9, G315.4−2.3, G084.2+0.8, and G127.1+0.5). We compare our limits with cooling curves for neutron stars and find that any putative neutron stars in these SNRs must be cooling faster than expected for traditional 1.35 M_⊙ neutron stars.