This phenomenon was demonstrated in a series of experiments performed by Justin Kruger and David Dunning and is now known as the Dunning-Kruger effect:

“The Dunning–Kruger effect is a cognitive bias in which “people reach erroneous conclusions and make unfortunate choices but their incompetence robs them of the metacognitive ability to realize it”. The unskilled therefore suffer from illusory superiority, rating their own ability as above average, much higher than in actuality; by contrast, the highly skilled underrate their abilities, suffering from illusory inferiority. This leads to a perverse result where less competent people will rate their own ability higher than more competent people. It also explains why actual competence may weaken self-confidence because competent individuals falsely assume that others have an equivalent understanding. “Thus, the miscalibration of the incompetent stems from an error about the self, whereas the miscalibration of the highly competent stems from an error about others.” (wikipedia)

They also

hypothesized that with a typical skill which humans may possess in greater or lesser degree […] incompetent individuals fail to recognize genuine skill in others” (wikipedia)

This is quite exactly the definition Errol Morris gave to “stupidity” a few days ago via his Twitter account:

“Apropos of nothing. My definition of a stupid person. A stupid person is a person who treats a smart person as though they’re stupid.” (Twitter)

UPDATE : looks like someone told Errol :

“Someone compared my definition of a stupid person to the Dunning-Kruger Effect, http://bit.ly/agqBDz. (I am deeply flattered.)” (Twitter)