This paper is concerned with the set-membership estimation problem for complex networks subject to unknown but bounded attacks. Adversaries are assumed to exist in the nonsecure communication channels from the nodes to the estimators. The transmitted measurements may be modified by an attack function with added noise that is determined by the adversary but unknown to the estimators. A novel set-membership estimation model against unknown but bounded attacks is presented. Two sufficient conditions are derived to guarantee the existence of the set-membership estimators for the cases that the attack functions are linear and nonlinear, respectively. Two strategies for the design of the set-membership estimator gains are presented. The effectiveness of the proposed estimator design method is verified by two simulation examples.