Most astronomers now accept that stars more massive than about 9 M 0 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

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... a level of one tenth of that of the Cas A central source, l-x~10 31 ergs S-l, 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 MG) neutron stars.