  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 $J_1$ and next-nearest-neighbor $J_2$ interactions ($J_2/J_1
\approx 0.5$). The double perovskites Sr$_2$CuTeO$_6$ and Sr$_2$CuWO$_6$ are
isostructural spin-1/2 square-lattice antiferromagnets with N\'eel ($J_1$
dominates) and columnar ($J_2$ dominates) magnetic order, respectively. Here we
characterize the full isostructural solid solution series
Sr$_2$Cu(Te$_{1-x}$W$_x$)O$_6$ ($0 \leq x \leq 1$) tunable from N\'eel 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 $x \approx$ 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 $J_2/J_1 \approx 0.5$ 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.
