Instrument failure isn't always detected by the crew. In the case of Air France 447 in 2009, even the instruments that were working correctly weren't properly understood.
I remember reading that - it is quite terrible but fascinating. It seemed pretty clear that the the crash was 100% human error due to the co-pilot basically panicking and losing his mind. Though it was initiated by an instrument failure.
The black box transcript from AF 447 is quite an interesting read: http://www.popularmechanics.com/technology/aviation/crashes/...