I rarely use IDEs, but compile-time checks are super useful for me during refactoring.
Sometimes I need to rename a class, a method name or a method signature that is used all over the program. I could use tests, but the compiler is much faster and much more comprehensive. So I change everything in one go and then only run the tests after that.
The speed in itself is great, and helps me get much faster feedback than I can get with tests. It's also great during explorational programming, or during the prototype phase: some errors can be detected during compilation, so I'm not greeted with an easily avoidable runtime error.
As for an example of a bug that would have been prevented by compile-time checking, I had one happen just last week. It was in a project with 100% test coverage, but we had undetected refactoring errors due to a lack of integration testing: there were (good) unit tests with mocks, but we missed them during refactoring, triggering a runtime error in production. We fixed that by writing a few simple integration tests and fixed the code. This happened just last week. No biggie, but would be avoided by a compiler.