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We compute bounds and forecasts on screened modified gravity theories, specialising to the chameleon model in Casimir force experiments. In particular, we investigate the classical interaction between a plate and sphere subject to a screened interaction of the chameleon type. We compare numerical simulations of the field profile and the classical pressure exerted on the sphere to analytical approximations for these non-linear field theories. In particular, we focus on the proximity force approximation (PFA) and show that, within the range of sphere sizes $R$ and plate-sphere distance $D$ simulated numerically, the PFA does not reproduce the numerical results. This differs from the case of linear field theories such as Newtonian gravity and a Yukawa model where the PFA coincides with the exact results. We show that for chameleon theories, the screening factor approximation (SFA) whereby the sphere is modelled as a screened sphere embedded in the external field due to the plates, fares better and can be used in the regime $D\gtrsim R$ to extract constraints and forecasts from existing and forthcoming data. In particular, we forecast that future Casimir experiments would corroborate the closing of the parameter space for the simplest of chameleon models at the dark energy scale.