Techniques are described for providing managed computer networks. In some situations, the techniques include managing communications for computing nodes of a managed computer network by using one or more particular computing nodes of the managed computer network that are configured to operate as intermediate destinations to handle at least some communications that are sent by and/or directed to one or more other computing nodes of the managed computer network. In addition, the techniques may include managing the communications in accordance with configured failure behavior specified for one or more computing nodes of the computer network, such as specified failure behavior for a computing node configured to operate as an intermediate destination that indicates how communications that would otherwise be routed via the intermediate destination computing node are to be handled if the intermediate destination computing node fails or is otherwise unavailable (e.g., to block or allow such communications).