A theoretical class of nonlinear systems is considered. An n-dimensional pipeline system is examined, as an example system, for investigating the effectiveness of a high-gain observer-based residual for detecting faults. Detailed conditions are given for the investigation of residual effectiveness, such as selection of operating points, control inputs, sensor faults, component faults, fault positions and observer eigenvalues. Then, for various values of n, defined fault types and defined fault positions, the observer-residual performance is noted and characteristics are derived and compared. Qualitative and quantitative evidence in graphical and table form shows, in several ways, how the performance and effectiveness of observer-based residuals change as n, and system complexity, increases.