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This paper is the first in a series of three, the remaining two of which will appear at a later date. Performance bounds are derived in this series of papers for causal state estimation and regulation problems employing mean-square criteria. The systems considered are partially observed finite-dimensional continuous-time stochastic processes driven by additive white Gaussian noise. The particular systems to which the bounds apply are those which, when modeled by Ito differential equations, contain drift (.dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. The bounds derived in the complete series of papers include both upper and lower bounds for causal state estimation and for a quadratic regulator problem. The upper bounds pertain to the performance of some simple, almost linear designs reminiscent of the designs which are optimal in the linear case. The lower bounds pertain to the optimum performance attainable within a broad class of candidate designs. This paper treats the upper bound for estimation. The second paper presents the estimation lower bound. The estimation performance bounds are independent of the control law, giving a particularly simple form to the control performance bounds which are developed in the third, and final, paper. All the bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.