  We explore the interplay between randomness and magnetic frustration in the
series of $S = 1/2$ Heisenberg square-lattice compounds
Sr$_2$CuTe$_{1-x}$W$_x$O$_6$. 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
Sr$_2$CuTe$_{1-x}$W$_x$O$_6$ with frustration being less significant.
