I've worked at two companies whose clients have all been non-profits - this has put me in a position to witness the state of dozens of them. I've also spent just as much time in the private sector, which has lent some contrast.
While there are definitely some very well-run non-profits, they are (in my experience) more the exception than the rule. Having said that, I wouldn't say the majority of private sector companies are well-run either.
But in general, the level of flakiness, obsessive behaviour, and eccentricness bordering on insanity was much, much higher at the non-profits. I don't necessarily think that it translated into poor performance as a rule, but at the very least it isn't pretty.
The office parties are a hell of a lot more fun, however, and for my part I'd prefer the company of someone at a non-profit. I have a theory that there's just as much crazy behaviour at for-profits, it's just less tolerated and therefore better hidden (rather than absent).
But to state that one isn't aware of the existence of large, decades-old, indisputably functioning non-profit organizations (like e.g. Médecins Sans Frontières) seems, well... disconnected from reality.
I wouldn't call any of the non-profits I've worked at failures.
Hell, many people have described companies like Microsoft as badly disfunctional, but it would be hard to argue that they are not successful.
If I stepped on anyone's toes with my post, I apologize. I was not trying to take a swipe at any one or any type of organization. I'm still involved with a number of organizations and admire and respect the work that they do.
I'm not trying to make a general case that there is some specific quality of non-profits that makes it impossible for them to be functional.
As I said pretty clearly above, the wikimedia story sounds typical, based on my interactions with non-profits.