We explore the interplay between randomness and magnetic frustration in the series of S = 12 Heisenberg square-lattice compounds Sr2CuTe1???xWxO6. Substituting W for Te alters the magnetic interactions dramatically, from strongly nearest-neighbor to next-nearest-neighbor antiferromagnetic coupling. We perform neutron scattering measurements to probe the magnetic ground state and excitations over a range of x. We propose a bond-disorder model that reproduces ground states with only short-ranged spin correlations in the mixed compounds. The calculated neutron diffraction patterns and powder spectra agree well with the measured data and allow detailed predictions for future measurements. We conclude that quenched randomness plays the major role in defining the physics of Sr2CuTe1???xWxO6 with frustration being less significant.