The concentration dependence of the sedimentation rate has been studied for a variety of colloids with a magnetic core and nonmagnetic shell. The nonmagnetic shell was either a silica layer or, in the case of magnetic fluids, a surfactant layer. It was found that the (linear) concentration dependence is very much affected by the thickness of the nonmagnetic shell and varied from negative for repulsive particles to positive for a sufficiently strong net attraction. This change in sign is in accordance with a recent theory for sedimentation of dipolar hard spheres. The sensitivity of concentration-dependent sedimentation to interaction details is also illustrated by the significant influence of any van der Waals attraction and surface charge, which competes with the effect of magnetic attraction. Our study shows that surface charge may be important in nonaqueous magnetic fluids.