For debugging purposes, yes, but in production the original exception is more important. Note that AnotherException is also logged at warn level which is intended to assist with debugging.
Either way this is a confusing situation that you need to try and avoid (i.e. control exceptions in exception handling code more tightly). It is long standing, existing behavior that cannot change without causing regressions.
So you're actually raising the exception intentionally from the @ExceptionHandler method, and not just simulating an unexpected error during from an @ExceptionHandler. This is certainly not something that an @ExceptionHandler was designed for. I actually don't understand what you're trying to do. What do you want to happen to AnotherException, propagate to the Servlet container and become a 500?