One for all, or how to help testing many components of one system

I'm testing the systems cluster, contains the various components, located on the OSS (Operation Support Systems) level. These components provide the ability for Deutsche Telekom customers to  receive a variety of services (Internet, telephony, and others), from a technical point of view. The whole system, for various reasons, composed of many different components, many of which are tested separately under component testing phase.

 I want to share the experience of the interaction between the system and component testing stages. Tell what happens when the components are successfully tested, but there are errors in the interaction between them within the system. How is it possible to solve some problems in the context of this situation. And which decision we're trying to find in our project.