Usually the problem you'll get if you use a lot of functional/integration testing is that your test suite will run too slowly for you to run it often enough to validate your design. It ends up being something you run when you think you've completed a feature to make sure it works, but not after you've extracted a class to make your code cleaner.

I have also had times when I moved away from unit testing in favor of higher-level tests, but ultimately found that the quality of the code got worse and the build time became unacceptable. I'm not trying to generalize here, I'm only talking about my code on a particular project at a particular point in time.