Sometimes an ignoramus can out-do a know-it-all. The most well-known example of this is the recognition heuristic, whereby we choose an alternative that we recognize or are familiar with over another alternative that we don't recognize. This can work is, roughly speaking, when recognition is positively correlated with desirable consequences. A know-it-all who recognizes all of the alternatives has to rely on other things she knows about those alternatives, and if those things are not strongly correlated with desirable outcomes her choices may be worse than an ignoramus using the recognition heuristic. For a gentle introduction to this and related concepts, see my blog post. In a recent paper (Smithson, 2010) I proved that the recognition heuristic can out-perform knowledge-based cues even when recognition is not as strongly correlated with desirable outcomes as the knowledge cue is. The journal in which this paper appeared has numerous high-quality articles on this heuristic.
Another less well-known phenomenon along these lines is where an error-prone perceiver can nevertheless provide better information about the world than a perfectly accurate perceiver. See my blog post for an introduction to this idea.
Likewise, even some of our cognitive shortcomings such as Gambler's Fallacy actually earn their keep when we're not in casinos but making judgments about real-world dynamics. I presented an experimental demonstration that humans are good at short-range predictions of chaotic processes because of Gambler's Fallacy (Smithson, 1997). I have a blog post on this too.
References:
Smithson, M. (1997). Judgment under chaos. Or
Smithson, M. (2010) When less is more in the recognition heuristic.
Judgment and Decision Making, 5, 230-243.