In nearly all instances of flight one is flying a heading, and (if radio navigating) one uses navaid indications (including ones from GPS if applicable) to modify the heading. But one is still flying a heading.
With an instrument failure, the question becomes: how long it is before the duff instrument indication becomes apparent.
I suppose much depends on how "authentic" the failure is. On the dodgy VOR approach referred to, the failure was pretty damn authentic...