Recent photometric analyses of the colour-magnitude diagrams of young massive clusters (YMCs) have found evidence for splitting in the main sequence and extended main-sequence turn-offs, both of which have been suggested to be caused by stellar rotation. Comparison of the observed main-sequence splitting with models has led various authors to suggest a rather extreme stellar rotation distribution, with a minority (10-30 per cent) of stars with low rotational velocities and the remainder (70-90 per cent) of stars rotating near the critical rotation (i.e. near break-up). We test this hypothesis by searching for Be stars within two YMCs in the Large Magellanic Cloud (NGC 1850 and NGC 1856), which are thought to be critically rotating stars with decretion discs that are (partially) ionized by their host stars. In both clusters, we detect large populations of Be stars at the main-sequence turn-off (~30-60 per cent of stars), which supports previous suggestions of large populations of rapidly rotating stars within massive clusters.