A fundamental issue in analyzing diffusion in heterogeneous media is interpreting the space dependence of the associated diffusion coefficient. This reflects the well-known Ito-Stratonovich dilemma for continuous stochastic processes with multiplicative noise. In order to resolve this dilemma it is necessary to introduce additional constraints regarding the underlying physical system. Here we introduce a mechanism for generating nonlinear Brownian motion based on a form of temporal disorder. Motivated by switching processes in molecular biology, we consider a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast-switching limit one obtains Brownian motion with a space-dependent diffusivity. We show that the resulting multiplicative noise process is of the Ito form. In particular, we solve a first-passage time problem for finite switching rates and show that the mean first-passage time reduces to the Ito version in the fast-switching limit.