Unit tests don't find bugs. They find regressions. This is a painful lesson I learned when I first started doing TDD (test-driven development), and it's well known among most TDD circles.
TDD's goal is to prevent programmers from introducing new bugs into working code. However, when you're writing code from scratch, your tests won't help you find all the bugs in your code. That's because you can't possibly write tests for all the ways your software will be used (or abused). When I first started doing TDD, I had really good tests, but I was too tired to do much exploratory QA. However, my boss wasn't, and I was very embarrassed to find that my software had lots of bugs. Simply put, he used my software in ways that I hadn't intended.
I've seen a lot of companies that don't bother writing any tests or doing any QA. They just let their users find all the bugs. Needless to say, I've never had respect for those companies.
However, it's growing more and more popular to destaff the QA department and just require engineers to write lots and lots of tests. Often, these are in the form of unit tests. Even though integration tests can conceivably catch more bugs, they take much longer to run. Hence, even integration tests are often deprioritized.
What I'm discovering is that a lot of projects have both lots of unit tests and lots of bugs. These are bugs that could have been found manually by a QA engineer, but it seems that manual QA testing (i.e. exploratory testing) has gone out of vogue.
I used to think that code that was well-documented, well-styled, well-tested, and code reviewed would rarely have bugs. Sadly, I no longer believe that to be the case. I think we need to go back to the days when we had decently-sized QA departments, perhaps in addition to all the other things we do.
To tweak what Knuth said, "Beware of the above code. I have only tested that it works. I haven't actually tried it."