difference between failed and broken test
Solution 1:
I don't know about nUnit specifically but in general testing terminology, the two are well defined.
A failing test is one which is well written and which has determined that the component being tested is broken. This usually happens when an assertion fails in the test case.
A broken test is a badly written test that fails because of an error in itself rather than in the module being tested. This can, for example, be a syntax error in the test.
Solution 2:
A failed test is a test which violated the asserts. A broken test could not be run because it's broken.