What should I do if the admission committee has missed something in my application and rejected me? I applied for a graduate program and I was rejected, but there is evidence suggesting that there was a mistake in reviewing my application. Now the school seems reluctant to reconsider the application. What should I do? Is there any real chance of being reconsidered?

Here are some details about the story:

The reason for rejection was mentioned in my rejection letter, and apparently, it was a very objective and quantifiable admission requirement that they thought I don't meet. But that was simply wrong, and they seem to have misunderstood (or even completely missed) one of my documents. It is a university in a different country, and apparently, they misinterpreted the document, which was issued by my undergrad school, and missed important information. I explained the document to them and they responded with a change in the reason, but the new reason in a way contradicts the previous one and it again seems to be about missing the exact same information related to that document. The new reason is also even more objective and almost purely bureaucratic, so I explained the policy of my undergrad university to them and what the information in the document means, but then I did not hear back from them anymore. 

What should I do in this situation? So far I've just been explaining to them how there might have been some confusions, but with their most recent response, it seems to me that most probably, they misunderstood my documents, and might have even completely missed one of them! I'm afraid of pursuing this more seriously and directly because the department is one of my favorite departments and I like to keep the option of going there later in my academic career open.
I am assuming you don't have a prospective advisor at this institution working with you on your admission. If you do, they will be in a much better position than us to tell you how to navigate this situation!

The concrete policy would depend on the country or individual institution, but to me it seems unlikely that they would reverse such a decision after the fact. In all probability they have limited places that they have filled already. They might also have rejected you for reasons they don't like to say openly, such as because the other candidates were equally good and didn't require double-checking the paperwork.

To know more about your chances of appeal, you would have to look into how similar cases have been handled by this department in the past, but that might be quite difficult without a connection to someone with historical insider info. While you are contesting the decision, the administration will naturally try to say as little as possible to avoid giving you ammunition.

If re-applying next year is an option for you, I would try to strike a more collaborative tone and ask how you should present your documents so as to avoid confusion in the future. And also try to strike up some contact with a faculty member who might help you. They don't necessarily need to actually do much, but an internal interest will give the administration some incentive to spend time answering your questions.