In Game of Thrones there's a bank that "always gets its due". It lends to noblemen and kings but is entirely independent of their politics (it seems like). If one nobleman fails to pay them back they will back another who will take his place -- and their debts.
It just seems like an independent bank is a modern concept. A power like that historically would be taken over and become the "Royal Bank".
No, historically medieval/renaissance bankers always had an enormous amount of difficulties with lending to monarchs and had little recourse to them when faced with sovereign defaults. When faced with political/economic pressure from kings they usually folded.
The best example of this and the closet analogue to the Iron Bank is the Medici banking family of Florence. They were the largest and most respected bank of their time and had branches all over Europe. When Edward IV (the king Robert Baratheon is based on) proved unable to pay his loans, he was actually able to leverage the importance of the English wool trade to Florence to get the Medicis to -keep- lending him money. The Medicis also made a bunch of bad loans to Charles the Bold (who subsequently perished at the Battle of Nancy) which it was not able to recover.
Notably during the war of the Roses the Medicis lend too much to the Lancastrian side, which proved disastrous because of the death of their leaders in the war and neither the Yorkists nor the Tudors would pay those outstanding loans.
Eventually the Medicis collapsed because of the French invasion of Italy. They were evicted from Florence and had their assets confiscated.
The successors to the Medici, the Fuggers, didn't do better in this regard, the Spanish monarchy defaulted on them three times and at times outright seized their assets for funds, and there was little they could do about it except negotiate.
Pre-modern banking institutions were hideously vulnerable to sovereign power and one of the things retarding the development of the financial system was that many people would actively avoid banks because of the ease in which kings could simply confiscate their wealth.
Edit: Edward III was not actually indebted to and defaulted on the Medici, but to Peruzzi banking family, but I think the point still stands with regards to the weakness of the banking system under sovereign pressure.
Historically, the big motivator for leaders to pay back loans is the possibility of getting cut off from future loans otherwise.
Money buys soldiers, mercenaries, guards, and so-on. A leader who doesn't take out loans can be outbid by one who does. Falling behind in the money-loaning race gets you killed. If a leader stops paying his debts, he stops getting new loans, and he risk being overtaken.
There aren't many historical examples of entities fighting to get people to pay back loans, or even sponsoring other parties to get their money back. Its because they don't need to. An example of this would be when Mexico defaulted on its payments to Spain, Britain, and France.
Spain and Britain just stood back and did nothing, which was the best move. France chose to invade instead. After a long war (which created the holiday Cinco de Mayo), France had lost a catastrophic amount of money fighting. Spain and Britain, on the other hand, where eventually paid back and made a nice profit.
The Iron Bank's model of directly interfering in politics is probably a bad one.
Furthermore, there has never been one party that has a monopoly on banking. Loaning money doesn't take much; anyone with money (churches, lords, merchants, and banks) can do it.
So no, there isn't a historical basis for the Iron Bank.
Sources: The Dictator's Handbook
The French Intervention in Mexico (1862-67), Military History Online