An example that comes to mind immediately is of a child learning to ride a bike. At some point the training wheels have to come off, which in most cases means at least one or two skinned knees. While training wheels, riding on the back lawn, and having mom or dad jog along side the bike can move a child towards competent bike riding, there comes a time when further development isn't possible without a higher level of risk-taking. The same is true of learning a new language--books from the library, Rosetta Stone, and online tutorials will help, but to become proficient, I have to actually speak to other human beings and risk looking stupid. The good news is that eventually I'll figure it out and get better.
There are examples of this in more formal academic settings as well. First-year writing courses almost always employ peer reviews or writing groups where students read and critique one another's writing. While an argument could be made that the feedback students get from peers is often useless (e.g. hollow statements of "great job" or "you need a semicolon here."), the principle still holds true that the point is for students to make their writing public and get feedback--that means the risk of having someone tell you that you have been unclear, boring, etc. And, of course, scholarly work is frought with risk (submitting papers to peer-reviewed journals, sharing ideas at conferences, etc.). But, these "risky" behaviors lead to refined ideas and improved academic work. Our "service" assignments on campus can also lead to tremendous learning as pointed out by Gary Daynes in a recent blog post. Gary argues that it is those assignments that we are unprepared or slightly unqualified for that lead to the most meaningful learning and growth. But, again, there is the chance that we'll (at least initially) be viewed as ignorant or incompetent and that scares us (and those that make the assignments as Gary articulates very well).
Aside from my own thinking on this issue, I would argue that there is plenty of supporting evidence from the academic literature including Carol Dweck's work on intelligence (mindset), constructivist descriptions of learning that emphasize the role of social negotiation, and ideas from management literature regarding the role of transitional phases in organizational learning.
I'm convinced that most organizations don't take the right kinds of designed and thoughtful risks necessary for really great learning to occur.