Using a combination of high-resolution Hubble Space Telescope/WFPC2 observations, we explore the physical properties of the stellar populations in two intermediate-age star clusters in the Large Magellanic Cloud, NGC 1831 and NGC 1868, based on their color-magnitude diagrams. We show that both clusters exhibit extended main-sequence turn-offs. To explain the observations, we consider variations in helium abundance, binarity, age dispersions, and fast rotation of the clusters' member stars. The observed narrow main sequence excludes significant variations in helium abundance in both clusters. We first establish the clusters' main-sequence binary fractions using the bulk of the clusters' main-sequence stellar populations >1 mag below their turn-offs. The extent of the turn-off regions in color--magnitude space, corrected for the effects of binarity, implies that age spreads of order 300 Myr may be inferred for both clusters if the stellar distributions in color--magnitude space were entirely due to the presence of multiple populations characterized by an age range. Invoking rapid rotation of the population of cluster members characterized by a single age also allows us to match the observed data in detail. However, when taking into account the extent of the red clump in color--magnitude space, we encounter an apparent conflict for NGC 1831 between the age dispersion derived from that based on the extent of the main-sequence turn-off and that implied by the compact red clump. We therefore conclude that, for this cluster, variations in stellar rotation rate are preferred over an age dispersion. For NGC 1868, both models perform equally well.