This paper addresses the problem of leader-following consensus for networked multi-agent systems subject to limited communication resources and unknown-but-bounded process and measurement noise. First, a new distributed event-based communication mechanism on the basis of a time-varying threshold parameter is developed to schedule transmission of each sensor's measurement through a communication network so as to alleviate consecutive occupancy of communication resources. Second, a novel concept of set-membership leader-following consensus is put forward, through which the true states of all followers are guaranteed to always reside in a bounding ellipsoidal set of the leader's state. Third, in the case that full information of followers’ states are not measurable, a distributed observer-based consensus protocol is presented to provide a set-membership estimation of each follower's state. Then, based on a recursive computation of confidence state estimation ellipsoids and leader state ellipsoid, a delicate convex optimization algorithm in terms of recursive linear matrix inequalities is proposed to design desired consensus protocol and event-based mechanism. Finally, an illustrative example is given to show the effectiveness and advantage of the developed approach.