For a class of nonlinear systems subject to disturbances, an observer is proposed to estimate time-varying intervals in which their state variables are guaranteed to stay all the time. The objective is to effectively deal with nonlinearities depending on unmeasured variables which have been usually treated as uncertainty in observer design. Focusing on nonlinearities in a polytopic form, this paper shows how an interval observer can replace nonlinearities in unmeasured variables by nonlinearities in estimated intervals. Theoretical guarantees and simulation comparisons are presented to demonstrate that the use of interval-dependent nonlinearities gives better estimates than the use of an overbounding observer.