We consider a class of nonlinear feedback control systems with monotone nonlinearities and several stationary states. If the system is under an almost periodic perturbation, one can obtain conditions for existence of almost periodic oscillations. Our purpose is to estimate the fractal dimension of the trajectory closure of forced almost periodic oscillations obtained by the mentioned way. We show that within the result of I. M. Burkin and V. A. Yakubovich, which extends the result of M. A. Krasnoselskii et al. on the existence of exactly two almost periodic solutions (the stable one and the unstable one) in the case of two stationary states, it is possible to obtain some estimates of the fractal dimension. This estimate depends on some properties of Diophantine approximations for the frequencies of the almost periodic perturbation. We also apply a similar approach to study almost periodic oscillations in the perturbed Chua circuit, where the unperturbed system has three stationary states. We provide some analytical upper estimates of the fractal dimension and some numerical simulations conrming that upper estimates provided can be exact.