I've written, and seen plenty of Integration tests as described by Gus - to test the entire vertical slice of software in a realistic environment without stubs, mocks or fakes as much as possible. I find those very useful because programmers tend to surround themselves with stubs, mocks and fakes forgetting about the real world only to have it turn up in the production environment.
What I've never seen is anyone writing integrated tests attempting to test the minute interactions of every component with every neighbour. Doing so would be madness as you've pointed out.