We present deep color-magnitude diagrams for three rich intermediate-age star clusters in the LMC, constructed from archival ACS F435W and F814W imaging. All three clusters exhibit clear evidence for peculiar main-sequence turn-offs. NGC 1846 and 1806 each possess two distinct turn-off branches, while the turn-off for NGC 1783 shows a much larger spread in color than can be explained by the photometric uncertainties. We demonstrate that although all three clusters contain significant populations of unresolved binary stars, these cannot be the underlying cause of the observed turn-off morphologies. The simplest explanation is that each cluster is composed of at least two different stellar populations with very similar metal abundances but ages separated by up to ≈ 300 Myr. The origin of these unusual properties remains unidentified; however, the fact that at least three massive clusters containing multiple stellar populations are now known in the LMC suggests a potentially significant formation channel.