We study the thermodynamic and high-magnetic-field properties of the magnetic insulator Ba5CuIr3O12, 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 T, 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 are consistent with the power-law form P(J) ∼ J−α with α ≈ 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.