This paper derives tight performance upper and lower bounds on the downlink outage efficiency of K-tier heterogeneous cellular networks (HCNs) for general signal propagation models with Poisson distributed base stations in each tier. In particular, the proposed approach to analyze the outage metrics in a K-tier HCN allows for the use of general bounded path-loss functions and random fading processes of general distributions. Considering two specific base station (BS) association policies, it is shown that the derived performance bounds track the actual outage metrics reasonably well for a wide range of BS densities, with the gap among them becoming negligibly small for denser HCN deployments. A simulation study is also performed for 2-tier and 3-tier HCN scenarios to illustrate the closeness of the derived bounds to the actual outage performance with various selections of the HCN parameters.