Confidence in units != confidence in your production system. I've observed that overconfidence in "units" tends to blind devs to other potential problem areas.
The most egregious example was an application release that wouldn't even start - there was an obvious bug in main that would have been caught by simply running the damn program and using your eyeballs. But the unit tests didn't catch it because main was tested in a mocked environment. Oops.
Likewise, I've seen similar regressions in quality control, user experience, performance, etc. - we know these things are critical to the real-world success of software, yet unit tests don't touch them.
There are software developers who are so overconfident in their unit tests that they ship code without ever bothering to run the application! Functional requirements be damned. "But the unit tests said it worked..." is not a professional excuse. Don't be that developer. Edit: to be clear, this is a message to myself as well. I was that developer a few times!