Less than half of the SNRs within 5 kpc have identified central sources, and only three are identified as the remnants of Type Ia SNe. Here, we report a systematic effort to search for compact central sources in the remaining 23 SNRs of this distance limited sample. In this first paper we report Chandra observations of four SNRs. We have undertaken a systematic optical/IR identification program of the X-ray sources detected in the field of each SNR. Foreground and background sources have identifiable IR/optical counterparts, while the counterparts of neutron stars (or black holes) are expected to be very faint. We are able to account for all the well-detected X-ray sources and thus 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 <~ 10e31 ergs/s. 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 Mo neutron stars and that any putative pulsar must have low spin-down luminosities, Edot <~ 1e34 ergs/s. However, our limits are unable to constrain the presence or absence of more unusual options, such as relatively more massive neutron stars, neutron stars with exotic interiors, or quiescent black holes. In subsequent papers, we will report on the X-ray and optical/IR observations of the remaining members of the 5-kpc sample. Comment: 52 pages, 37 figures. Accepted for publication by ApJS. Uses emulateapj5.sty, onecolfloat5.sty