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We present a new partition-based fault-diagnosis technique for identifying error-capturing scan cells in a scan-BIST environment. This approach relies on a two-step scan chain partitioning scheme. In the first step, an interval-based partitioning scheme is used to generate a small number of partitions, where each element of a partition consists of a set of consecutive scan cells. In the second step, additional partitions are created using a previously proposed random-selection partitioning method. Two-step partitioning provides higher diagnostic resolution than previous schemes that rely either on random-selection partitioning or deterministic partitioning. We show via experiments that the proposed method requires only a small amount of additional hardware. The proposed scheme is especially suitable for a system-on-chip (SOC) composed of multiple embedded cores, where test access is provided by means of a TestRail that is threaded through the internal scan chains of the embedded cores. We present analytical results to characterize two-step partitioning, and present experimental results for the six largest ISCAS'89 benchmark circuits and two SOCs crafted from some of the ISCAS'89 circuits.