This paper focuses on consensus problems for high-order, linear multi-agent systems. Undirected communication topologies along with a fixed and uniform communication time delay are taken into account. This class of problems has been widely studied in the literature, but there are still gaps concerning the exact stability bounds in the domain of the delays. The novelty of this paper lies in the determination of an exact and explicit delay bound for consensus. This is done in a very efficient manner by using the cluster treatment of characteristic roots (CTCR) paradigm. Before the stability analysis, a state transformation is performed to decouple the system and simplify the problem. CTCR is then deployed to the individual subsystems to obtain the stability margin in the domain of the delays without the conservatism introduced by other approaches more frequently found in the literature. Simulation results are presented to support the analytical claims.