They are often sufficient for a great deal of projects. If all it takes to convince you it's "good enough," are a handful of examples then that's it. As much as you need and no less.
However I find we programmers tend to be a dogmatic bunch and many of us out there like to cling to our favoured practices and tools. Unit tests aren't the only testing method. Integration tests are fine. Some times testing is not sufficient: you need proof. Static types are great but fast-and-loose reasoning is also useful and so you still need a few tests.
What's important is that we sit down to think about specifying what it means for our programs to be, "correct." Because when someone asks, "is it correct?" You need to as, "with respect to what?" If all you have are some hastily written notes from a bunch of meetings and long-lost whiteboard sessions... then you don't really have an answer. Any behaviour is, "correct," if you haven't specified what it should be.