Example methods and apparatus to monitor border gateway protocol sessions are disclosed. A disclosed example method includes detecting a failure of a first BGP session, initiating a sustained-down timer and a reset-timer in response to detecting the failure, identifying a sustained-down condition in response to the sustained-down timer exceeding a first time threshold, identifying a flapping condition in response to counting a threshold number of BGP failures and corresponding BGP re-establishments during a second time threshold of the reset-timer, and identifying a continuous flapping condition in response to detecting the flapping condition consecutively for a threshold number of instances.