Alice has a bank account number, but has forgotten which bank it is for. There are 4 banks, run by Bob, Carlos, David, and Eve.
She could find out by going to all of the banks and asking if they have the account number. However, if Eve learns Alice's account number, then Eve will go to Alice's actual bank and steal all of Alice's money.
Alice could hash the bank account number, and ask about the hash, but since the account number is only 8 digits, Eve could bruteforce the hash anyway. Then, Eve will go to Alice's bank and steal all of her money.
Alice could use a Zero Knowledge Proving Protocol, but how would the bank know which account number to check against without repeating the ZKPP for every account number? Each of them has thousands of customers.
Context: I'm writing a program to retrieve an encrypted copy of one's keys and friends-list for a project called Tox (http://wiki.tox.im/index.php/Proposal:Friendslist_Server). I'd like to make it automatically detect which server has it.