In this paper, we study fundamental properties of minimum inter-event times for several event-triggered control architectures, both in the absence and presence of external disturbances and/or measurement noise. This analysis reveals, amongst others, that for several popular event-triggering mechanisms no positive minimum inter-event time can be guaranteed in the presence of arbitrary small external disturbances or measurement noise. This clearly shows that it is essential to include the effects of external disturbances and measurement noise in the analysis of the computation/communication properties of event-triggered control systems. In fact, this paper also identifies event-triggering mechanisms that do exhibit these important event-separation properties.