Authentication protocols are designed to work correctly in the presence of an adversary that can prompt honest principals to engage in an unbounded number of concurrent executions of the protocol. This paper establishes a bound on the number of protocol executions that could be useful in attacks. The bound applies to a large class of protocols, which contains versions of some well-known authentication protocols, including the Yahalom, Otway-Rees, and Needham-Schroeder-Lowe protocols.