We present far-red, intermediate resolution spectroscopy of 572 photometrically selected, low-mass stars (0.2<M/M_sun<0.7) in the young open cluster NGC 2516, using the FLAMES spectrograph at the Very Large Telescope. Precise radial velocities confirm membership for 210 stars that have published rotation periods from spot-modulated light curves and for another 144 stars in which periodic modulation could not be found. The two sub-samples are compared and no significant differences are found between their positions in colour-magnitude diagrams, the distribution of their projected equatorial velocities or their levels of chromospheric activity. We rule out differing observational sensitivity as an explanation and conclude that otherwise similar objects, with equally high levels of chromospheric activity, do not exhibit spot-induced light curve modulation because their significant spot coverage is highly axisymmetric. We propose that the spot coverage consists of large numbers of small, dark spots with diameters of about 2 degrees. This explains why about half of cluster members do not exhibit rotationally modulated light curves and why the light curve amplitudes of those that do have mean values of only 0.01-0.02 mag.