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Variance component estimation algorithms, in conjunction with automated in-process measurement technology, can be effective tools for identifying and eliminating major sources of manufacturing variation in assembly processes. Whether a particular set of variation sources are diagnosable depends critically on how the sensor system is laid out. Diagnosability tests are mathematical in nature and provide little insight into why a particular sensor layout may be nondiagnosable or how to modify the layout to ensure diagnosability. This paper translates the mathematical diagnosability conditions into a set of more conceptually meaningful conditions that provide better insight into the reasons behind the nondiagnosability. Note to Practitioners - This paper was motivated by the problem of identifying and eliminating major sources of variation in discrete-part manufacturing, which are critical steps in improving product quality. The effectiveness of statistical algorithms for estimating sources of variation depends on whether the sensor system for measuring key product and process variables is laid out properly, so that a particular set of diagnosability conditions are satisfied. This paper translates the rather abstract mathematical conditions for diagnosability into a set of more intuitive and conceptually meaningful conditions. This provides practitioners with insight into why a sensor system may be nondiagnosable and how to add or adjust sensors in order to ensure diagnosability. The diagnosability characterization can also be used to enhance performance and reduce computational expense in numerical search strategies for optimizing sensor layout.