What causes the phenomenon that pilots don't believe their instruments, while they're working well?
Human decision making is an entire field of psychological research and there are many, many studies and books about it. Yet for all that we really don't know exactly why people make the kinds of mistakes they do, so I will summarize the theoretical work that I personally have studied, which are Heuristics and Cognitive Biases. 

The theory is that human brains simplify decision making by developing Heuristics, which are essentially algorithms which allow for quick and efficient information processing. Think of it this way; you don't have time to analyze every situation from the ground up. Imagine if every time you walked up to the cashier in a store, went to cross a street, or from a piloting perspective became too close to another airplane you had to examine it from first principles! It would be slow, exhausting, and possibly dangerous. Heuristics are the brain's sophisticated way to shortcut that so that decisions can be made fast, they are developed through personal experience, second-hand experience, and training. 

The problem is that heuristics are prone to biases and errors of logic. Heuristics can be developed which get to the wrong result with a given set of information, these lead people to making the wrong decisions, interpreting situations incorrectly when thinking about it anew would give an entirely different result. Ambiguous data can lead to the wrong heuristic being used, several options being available forces the brain to choose the best one. 

I think heuristics theory goes a long way to explaining some of the common mistakes that are made in the cockpit, this is from my study of the material and from time in the cockpit, what is not that well understood is how stress and motivational factors can influence them. There's also the role of biases, which cause deviations from logic and good decision making. Some relevant ones are:


Selection bias, where someone cherry-picks what they want to see so it fits their view
Confirmation bias, where someone ignores things that don't fit their view, this is also known as expectation bias
Attention Tunneling, where someone becomes fixated on one thing and ignores others


My personal view on this is that the human brain has a tendency to force heuristics to fit input when it is ambiguous, especially in busy or stressful situations. This can be a good thing as it makes people act when they might otherwise not have, but people tend not to notice when the brain's developed ways of interpreting information don't fit the context or arrive at an obvious error. We won't like the feeling of being wrong or we are busy with other things, so we don't re-evaluate information. Sometimes people die because of that. 

This is why airlines have re-curring situational training, and why that training is so incredibly important for safety. They train pilots to match what they see and feel to sets of actions so that the right decisions are made quickly.

EDIT: 
There was a comment asking whether these effects could be contagious, the answer is yes, sometimes incorrect ideas get passed along, and if they don't get challenged it can lead to disaster. Some of these effects are:


Availability bias, where an event happening to one person makes another think it is more likely to happen to them. One pilot's instrument failure makes another think theirs did too, even though it's not true
Authority bias, where someone believes something is true when the source is a famous or authoritative figure. A captain telling a FO that an instrument has failed makes the FO believe it because they're the captain
Groupthink, where social pressures to conform or avoid dis-harmony make people buy into bad decisions