A curious series of events allegedly led to two men’s encounter with National Security Agency police at the Fort Meade army base near Baltimore on Monday—an encounter that ended with the death of one and the hospitalization of the other.

Police statements Tuesday may point to why authorities quickly dismissed terrorism in the incident. It took place near the headquarters for the NSA—a U.S. intelligence agency charged with monitoring communications and collecting data—which is housed at the sprawling military installation....