Anything diagnostic has to do with disease, especially figuring out which one a patient has.
If your doctor wants to run diagnostic tests, then she isn't exactly sure what's wrong with you — so the tests will help diagnose you (figure out what's wrong). The TV show House is all about diagnostic tests: it features illnesses that are unusual and hard to determine. But anything diagnostic is trying to figure out what's wrong. A writing test could be called diagnostic if it's trying to determine how good or bad your writing skills are.
