Hardware duplication techniques are widely used for concurrent error detection in dependable systems to ensure high availability and data integrity. These techniques are vulnerable to common-mode failures (CMFs). Use of duplex systems with diverse implementations of the two modules has been proposed in the past for protection against CMFs. In this paper, we define a category of faults, called non-self-testable faults that undermine the data integrity of dependable systems. These faults produce identical errors at the outputs of the two modules of a duplex system and can potentially be caused by CMFs.The main contributions of this paper are: (1) techniques that identify non-self-testable faults in duplex systems, and (2) design methods that reduce the number of non-self-testable faults by test point insertion. We show that our algorithm for identifying non-self-testable faults runs orders of magnitude faster than exact techniques with minimal loss of accuracy. Also, there is a significant reduction in the number of test points required for duplex systems with diverse implementations compared to duplex systems with identical implementations. Thus, we can detect common-mode failures in diverse duplex systems using very few test points. These results are especially useful for systems with user-programmable logic elements that enhance the practicality of using diverse designs in duplex systems.