how is the money the FDIC has collected
Fees collected from the banks:
The FDIC receives no Congressional appropriations – it is funded by premiums that banks and thrift institutions pay for deposit insurance coverage and from earnings on investments in U.S. Treasury securities.
They also use the proceeds from liquidating the assets of failed banks to make payouts.
Are there country specific agencies with a similar mission?
Canada Deposit Insurance Corporation
Instituto para la Ptotección al Ahorro Bancario (Mexico)
Financial Services Compensation Scheme (UK) not quite like the FDIC
You'll have to search for others yourself. :)
Most importantly, are there any examples of a similar system that has failed?
As the Mythbusters say, "failure is always an option."
There is a statement on FDIC's website to the effect that they are backed by the "full faith and credit" of the U.S. government. That said, the FDIC maintains its own fund to make insurance payouts. Granted, in the shakiness of the 2008-2009 financial crisis they did start waving a red flag about their realistic ability to cover their obligation. Practically speaking, the government will likely step in if necessary.
This 2008 article regarding a propsed revamp of the UK's FSCS should be of general interest to you on this topic, though it does not answer the question of failed systems. (Well, as far as I know. I have only skimmed the article.)