Keep in mind that (IIRC) on the 737 (and probably most other aircraft) there are multiple pitot tubes.

Prior to digital flight management systems, there were two independent pitot tube sensors driving independent altimeters and airspeed indicators. One system drove the Captain's instruments, the other system drove the First Officer's instruments. And, in some cases a third independent system drove the instruments for the flight engineer.

Therefore, you could have one system fail, and still have a secondary system working. However, it was up to the flight crew to determine which system was correct (or incorrect).

With today's digital flight management computers, there are usually (in the Honeywell systems, such as EPIC) three flight computers reading in data from three independent sensors (pitot tubes included). Each computer checks itself against the other three. If one (or more) do not match with the others (within some tolerance), the three computers independently recalculate again until they match (within some tolerance). If two "match" and one if off, the other two have the "authority" to determine that the third computer is in a fault state and they "ignore" the calculations of the third computer and report the fault to the crew.

In a nutshell, there's multiple pitot tube systems on the aircraft. If one is "bad" you still have a back-up. The only rub is if the Crew (or flight computers) determines the wrong system to be the inaccurate one.