A systematic design methodology for state observers for a large class of nonlinear systems with bounded exogenous inputs (disturbance inputs and sensor noise) is proposed. The nonlinearities under consideration are characterized by an incremental quadratic constraint parameterized by a set of multiplier matrices. Linear matrix inequalities are developed to construct observer gains, which ensure that a performance output based on the state estimation error satisfies a prescribed degree of accuracy. Furthermore, conditions guaranteeing estimation of the unknown inputs to arbitrary degrees of accuracy are provided. The proposed scheme is illustrated with a numerical example, which does not satisfy the so-called "matching conditions."