The 'criticism-weakness-failure' essay is common in MBA admissions essays because it is a test of an applicant's maturity, self-knowledge, honesty, and ability to learn from mistakes. It is, in other words, the biggest indicator of real leadership ability and potential.
Sample questions are:
Tuck 3. Discuss the most difficult constructive criticism or feedback you have received. How did you address it? What have you learned from it?
Wharton 3. Describe a failure that you have experienced. What role did you play, and what did you learn about yourself?
HBS 2. What have you learned from a mistake?
Columbia 3. Please provide an example of a team failure of which you've been a part. If given a second chance, what would you do differently?
Applicants to business school very often struggle with these essays because they feel that admitting a weaknesses or sharing a time when they failed erodes their candidacy. In fact, it does just the opposite. Leaders know their weaknesses, and can admit them to themselves and others in order to work on them, or work around them. It shows self-insight and points to seniority. No one is comfortable talking about their weak spots and failure. But nobody is perfect or has not failed. Not Bill Gates, not Richard Branson, not me, nor you, nor the admissions officer.
So it is not admitting a weakness is what will get you marked down, because it's like waving red beacon that betrays inexperience and a junior mindset. If you 'have no weaknesses' that just tells the admissions committee that you don't know what they are yet or that you're too immature to face them. It says you don't know yourself, therefore you don't yet know where you will mess up. You are a liability to yourself and your company.
Take a tip from George Soros, self-made billionaire, philosopher, philanthropist, social reformer, and fund manager extraordinaire - famous for 'breaking the Bank (of England)' by shorting the pound sterling in 1992 - who shares this candid account of his weaknesses:
I'm a very bad judge of character. I'm a good judge of stocks, and I have a reasonably good perspective on history. But I am, really, quite awful in judging character, and so I've made many mistakes. It took me five years and a lot of painful experiences to find the right management team. I am please that finally I found it, but I cannot claim to be as successful in picking a team as I have been in actually managing money. I think that I'm very good as a senior partner, or boss, because I have a lot of sympathy for the difficulties that fund managers face. When they are in trouble I can give them a lot of support, and that, I think, has contributed toward creating a good atmosphere in the firm. But I'm not so good at choosing them. Soros on Soros: Staying Ahead of the Curve, Wiley & Sons, NY, 1995, p.18
See, the greatest business leaders all have weaknesses and all have made significant mistakes in their careers and their lives. And note his tone - Soros is candid, straightforward, and objective in his self-analysis. He shares measured self-insight with the reader. He doesn't try to slip in softening or deflecting phrases, or hide behind humor; nor is he self-excusing or whining and looking to blame others, the mark of a too-junior applicant. The point is not to prove that you don't fail, or won't fail. It is to prove that you have the insight into yourself to be able to recognize and compensate for your weaknesses.
What Adcom wants to know is not how you avoided failure, but how you managed it, what you learned, what insights into yourself you gained, and how you grew from there. They want to see that you have the will and the insight to locate and understand the source of your mess-up - the underlying weaknesses that caused it - and that you have the maturity to face and work on the issue.
To summarize: the weakness / failure essay is not testing to see if you have weaknesses. We all do. It is a test of your self-knowledge and maturity. The committee wants to see if you can candidly face, discuss, and work on your flaws, or if you will you try to hide them or blame circumstances or other people. This is a significant test of your readiness for senior leadership.