Mock things like database access, or better yet pass the database connection via an interface and create a test version of the database access so you can create a database-like thing for your tests (e.g. an in-memory database).

If you are depending on another component of the application (a lexer, a JSON class, a maths function, etc.) don't mock that because if that class breaks you want tests to fail instead of silently passing because the broken class/function was mocked.

If you are depending on thirdparty libraries, don't mock those unless you have to (i.e. if you cannot run the tests). This will help avoid unexpected bugs after upgrading libraries or if supporting different versions of a library.

If you are writing code for a complex infrastructure (e.g. an plugin for an IDE), try to use test-specific versions of enough of the infrastructure to get the rest functioning, and use the real versions of as much as you can. This will help pick up issues in your code when new versions of that infrastructure make changes -- you want your tests to fail in this case, as the real code would fail.

Understand why your tests are breaking and address that. Use @Ignore as a last resort. If APIs or behaviour has changed, update the code and tests to reflect that. If you are supporting different versions, create version-specific compatibility layers.