Alistair_Heaton and TehMightyEngineer have both touched on an important point when it comes to the pilot/machine interface for diagnosing problems in the stress of an emergency. Some instrumentation, at least with regard to the signals/data, are closed loop systems. For example, GPS. The GPS satellites contain data streams that works with the on-board receivers and the mix of various GPS satellites being used, typically 4-5 satellites for the heavy iron on which I have most of my experience. The GPS receivers on-board and the satellites in the constellation being used can talk back and forth to detect, isolate, ignore and/or call up redundant satellites without pilot intervention. Thus the data presented to the pilot is reliable unless a flag and/or warning notifies the pilot not to use the data.
Air data systems are (at least as far as the data provided to the pilot) not closed loop systems. What I mean is there is no satellite constellation, ground station, airborne station or any other system in place outside and independent of the aircraft against which the air data system can compare what it THINKS is correct, and thus no feedback outside the aircraft itself to say to the air data system, "Whoa, the data you are feeding to the pilot sucks, throw a flag!"
Thus the air data systems can really only compare against each other. On some aircraft there are actually four air data systems: Pilot, CoPilot, Standby and Alternate. This gives a lot of redundancy unless the situation that is affecting one system affects all systems in a similar manner.
To illustrate: In a GPS constellation the probability (I don't have hard data in front of me, using relative terms) is very unlikely that more than one satellite will develop the same malfunction at the same time as they are thousands of miles apart and fully independent from each other. However, even on an aircraft with four air data systems, if all the pitot tubes and/or all of the static ports are affected in a similar manner by the same phenomenon (say for example rapid onset of icing at altitude), the various systems still agree with each other even though all of the systems are now lying to the pilots.
Of course physical separation on the exterior of the aircraft and other means (software processing) are used to try and provide as much independence as possible along with opportunities to find deviations that can provide warnings to the crew, but it is not as easy to provide independent comparisons between on-board systems as it is with signals and data originating outside the aircraft to use for cross check.
