  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.
