We test the behavior of the United States (US) West Coast ShakeAlert earthquake early warning (EEW) system during temporally close earthquake pairs to understand current performance and limitations. We consider performance metrics based on source parameter and ground‐motion forecast accuracy, as well as on alerting timeliness. We generate ground‐motion times series for synthesized earthquake sequences from real data by combining the signals from pairs of well‐recorded earthquakes (⁠4.4≤M≤7.1⁠) using time shifts ranging from −60 to +180 s. We examine fore‐ and aftershock sequences, near‐simultaneous events in different source regions, and simulated out‐of‐network and offshore earthquakes. We find that the operational ShakeAlert algorithms Earthquake Point‐source Integrated Code (EPIC) and Finite‐Fault Rupture Detector (FinDer) and the Propagation of Local Undamped Motion (PLUM) method perform largely as expected: EPIC provides the best source location estimates and is often fastest but can underestimate magnitudes or, in extreme cases, miss large earthquakes; FinDer provides real‐time line‐source models and unsaturated magnitude estimates for large earthquakes but currently cannot process concurrent events and may mislocate offshore earthquakes; PLUM identifies pockets of strong ground motion, but can overestimate alert areas. Implications for system performance are: (1) spatially and temporally close events are difficult to identify separately; (2) challenging scenarios with foreshocks that are close in space and time can lead to missed alerts for large earthquakes; and (3) in these situations the algorithms can often estimate ground motion better than source parameters. To improve EEW, our work suggests revisiting the current algorithm weighting in ShakeAlert, to continue developments that focus on using ground‐motion data to aggregate alerts from multiple algorithms, and to investigate methods to optimally leverage algorithm ground‐motion estimates. For testing and certification of EEW performance in ShakeAlert and other EEW systems where applicable, we also suggest that 25 of our 73 scenarios become part of the baseline data set.