I like unit tests to test the expected cases (positive or negative, as long as they're expected), to prove to myself that they conform to the spec. Then, if bugs are found (ie unexpected cases are discovered), I like to add tests for those, but I don't need to try and think of them up front. This way the unit tests will prevent regressions (both on spec conformance and bugs) as the code changes over time.

Integration tests are necessary for the exact reason you mention: just because things works in isolation, doesn't mean they will work together.

Ideally, for catching unknown bugs, I would also have a property-based generative test suite that is run over night against lots of random scenarios. This isn't always possible (in fact, so far, I've sadly only worked on one project where we did this, but I'd like this to become more common).