In this article, we consider the problem of parameter identification and state estimation of an uncertain continuous-time linearly parameterized nonlinear system with output dependent nonlinearities subject to exogenous disturbances. A set-based adaptive estimation is proposed in which the parameters and the states of the system are estimated along with an uncertainty set guaranteed to contain the true unknown values. The set-update approach is such that the sets are updated only when an improvement in the precision of the parameter estimates and the state estimates can be guaranteed. The formulation provides robustness to parameter estimation error and bounded disturbances. The adaptive estimation technique can be viewed as an adaptive interval observer. Simulation examples are used to illustrate the effectiveness of the developed procedure and ascertain the theoretical results.