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Core-collapse supernovae (SNe) expand into a medium created by winds from the pre-SN progenitor. The SN explosion and resulting shock wave(s) heat up the surrounding plasma, giving rise to thermal X-ray emission, which depends on the density of the emitting material. Tracking the variation of the X-ray luminosity over long periods of time thus allows for investigation of the kinematics of the SN shock waves, the structure of the surrounding medium, and the nature of the progenitor star. In this paper, X-ray observations of five of the oldest known X-ray SNe-SN 1970G, SN 1968D, SN 1959D, SN 1957D, and SN 1941C-are analyzed, with the aim of reconstructing their light curves over several decades. For those SNe for which we can extract multiepoch data, the X-ray luminosity appears to decline with time, although with large error bars. No increase in the X-ray emission from SN 1970G is found at later epochs, contrary to previous reports. All five SNe show X-ray luminosities that are of comparable magnitude. We compare the late-time X-ray luminosities of these SNe to those of supernova remnants (SNRs) in the Galaxy, which are a few hundred years old, and find that when the tentative decline is taken into account, the luminosity of the old SNe studied herein could fall below the luminosity of some of the younger SNRs within a few hundred years. However, the X-ray luminosity should begin to increase as the SNe expand in the Sedov phase, thus reaching that of the observed SNRs.