The spin-1/2 square-lattice Heisenberg model is predicted to have a quantum disordered ground state when magnetic frustration is maximized by competing nearest-neighbor J1 and next-nearest-neighbor J2 interactions (J2/J1≈0.5). The double perovskites Sr2CuTeO6 and Sr2CuWO6 are isostructural spin-1/2 square-lattice antiferromagnets with Néel (J1 dominates) and columnar (J2 dominates) magnetic order, respectively. Here we characterize the full isostructural solid-solution series Sr2Cu(Te1-xWx)O6(0≤x≤1) tunable from Néel order to quantum disorder to columnar order. A spin-liquid-like ground state was previously observed for the x=0.5 phase, but we show that the magnetic order is suppressed below 1.5 K in a much wider region of x≈0.1-0.6. This coincides with significant T-linear terms in the low-temperature specific heat. However, density-functional theory calculations predict most of the materials are not in the highly frustrated J2/J1≈0.5 region square-lattice Heisenberg model. Thus, a combination of both magnetic frustration and quenched disorder is the likely origin of the spin-liquid-like state in x=0.5.