It is well known that the thermal noise behavior at the terminals of any LTI RLC circuit can be predicted from knowledge of the driving-point impedance and temperature alone. This paper offers the conjecture that similar results hold if the capacitors and inductors are nonlinear. We refine the conjecture by analyzing the behavior of an RLC bridge circuit with the nonlinear inductor and capacitor carefully matched so the terminal behavior reduces to that of a linear resistor R. We show that the terminal noise current is precisely that predicted by the Nyquist-Johnson model for R if the driving voltage is zero or constant, but not if the driving voltage is time-dependent or the inductor and capacitor are time-varying. This paper makes exact calculations using techniques from stochastic differential equations and using reversibility arguments.