Recently, necessary and sufficient conditions for output synchronization of linear systems via diffusive couplings have been reported. In this paper, we study the case when such conditions are not satisfied and exact synchronization is impossible. In particular, we study two kinds of heterogeneous linear networks: (i) non-identical harmonic oscillators and (ii) double-integrators. We show that static diffusive couplings render heterogeneous networks of harmonic oscillators asymptotically stable. Networks of non-identical double-integrators, in contrast, are not asymptotically stable but synchronize with bounded synchronization error depending on the network topology and the heterogeneity in the agent dynamics. Numerical examples illustrate the results.