  We study the thermodynamic and high-magnetic-field properties of the magnetic
insulator Ba$_5$CuIr$_3$O$_{12}$, which shows no magnetic order down to 2 K
consistent with a spin liquid ground state. While the temperature dependence of
the magnetic susceptibility and the specific heat shows only weak
antiferromagnetic correlations, we find that the magnetization does not
saturate up to a field of 59 Tesla, leading to an apparent contradiction. We
demonstrate that the paradox can be resolved, and all of the experimental data
can be consistently described within the framework of random singlet states. We
demonstrate a generic procedure to derive the exchange coupling distribution
$P(J)$ from the magnetization measurements and use it to show that the
experimental data is consistent with the power-law form $P(J)\sim J^{-\alpha}$
with $\alpha \approx 0.6 $. Thus, we reveal that high-magnetic-field
measurements can be essential to discern quantum spin liquid candidates from
disorder dominated states that do not exhibit long-range order.
