Behavioral research demonstrates a critical transition in preschooler's mental-state understanding (i.e., theory of mind; ToM), revealed most starkly in performance on tasks about a character's false belief (e.g., about an object's location). Questions remain regarding the neural and cognitive processes differentiating children who pass versus fail behavioral false-belief tasks and the extent to which there is continuity versus change in the ToM neural network. To address these questions, we analyzed event-related spectral power in the electroencephalogram (EEG) to investigate how preschoolers' neural activity during passive viewing of false-belief scenarios related to their explicit behavioral ToM performance. We found that neural activity during passive viewing of false-belief events (6-9 Hz EEG 'alpha' suppression in right temporoparietal [RTP] electrodes) strongly related to children's explicit ToM. However, children's RTP alpha suppression differed depending on their explicit behavioral ToM performance: Children who did better on a broad battery of standard ToM tasks and who passed explicit behavioral false-belief tasks showed greater RTP alpha suppression when the character's belief first became false (during the 'location-change' event); whereas children who did worse on the ToM battery and who failed explicit behavioral false-belief tasks showed greater RTP alpha suppression only later when they could evaluate the character's behavior in the context of prior events (during the 'active-search' event). Findings shed light on what differentiates preschoolers who pass versus fail explicit false-belief tasks and raise questions about how to interpret existing neuroscience data from ToM tasks across infancy to adulthood.