We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N = 4 $$ \mathcal{N}=4 $$ superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.