Or if they are, they can become self-aware of their own ignorance by learning something about what they are ignorant of. On this point, Chris Lee has an interesting piece in Arstechnica about the Dunning-Kruger effect that ends with some advice that I have found helpful in dealing with my own ignorance (or lack of a certain skill) from time to time:
It is also important to confront people with their own failings. "There is also some thought that perhaps we should give people experience with their overconfidence," Dunning noted. "That is, get them to make an overconfident display, and then expose it for what it is, so that people are more on guard for such an issue. For example, in some areas, people learning to drive are exposed to horrible driving conditions, but not taught how to handle them. Instead, they are given enough frightening experience that they would never think to drive in icy or snowy conditions. I would not consider this a negative approach to education. As Anatole France said, a proper education isn’t what you know, it’s being able to separate what you know from what you don’t."