Probably the most famous instances of state bankruptcy in history belong to Spain during the reign of Philip II (1556-1598), which defaulted on its debts five times in 1557, 1560, 1569, 1575, and 1596. It should be noted, though, that true bankruptcy isn't strictly possible for states, since that's a status granted by a court to a debtor unable to meet payments; there wasn't a judge on earth in the 16th century who could bring the King of Spain to account. Rather, people use the term to describe what's more accurately called default, which is simply the act of not meeting payments on one's debts.

So why did Spain default so many times in the 16th century? Three words: war, war, and war! In the early modern period, the financial burden of waging war increased dramatically, as a result of the introduction of cannons (bronze is expensive!) and fortifications capable of resisting them; each angular bastion incorporated into the defenses of Amsterdam cost half a million florins, and the Dutch built 22 of them. Furthermore, the eclipse of heavy cavalry by infantry armed with pike and shot, made necessary by the need to constantly besiege powerful fortifications, increased the number of men on the state's payroll five or even tenfold, especially after the state took up the responsibility of providing their equipment and services (hospitals, religious care, retirement homes) for its soldiers.

At sea, the crown had to provide for tens of thousands of rowers on the royal galleys, and as the century progresses, naval artillery increasingly came to dominate war on the seas, with the victorious Christian fleet at Lepanto bringing some 1800 cannons to bear on the Ottomans. Beyond galley warfare on the Mediterranean, states began to shoulder more of the costs of building ships capable of weathering the turbulent Atlantic on voyages to and from the New World; armed with perhaps dozens of heavy anti ship guns, each was a serious investment, though Spain tended to rely on contracted merchant ships to ease the financial burden.

Philip II spent all but six months of his reign at war, and bore further debts from his father's campaigns across Europe and the Americas. To meet military challenges, states needed to be able to dramatically expand revenue in short order, which their tax collection systems weren't prepared for (though the Spanish managed to adapt, developing a system that allowed them to 'anticipate' tax revenue up to three years in advance); most of the crown's revenue came from sales taxes, along with highly volatile silver mining, which varied between 40% of the crown's income and almost zero, depending on the year. With unpredictable financial needs and highly variable revenue streams, the Spanish Habsburgs turned to borrowing to finance their wars. In response to short term crises that demanded immediate funds, they contracted asiendos, loans issued against the general credit of the king. While juros, which included lifetime annuities and perpetual bonds, concerned more long term borrowing; the latter gave the lender rights to a certain portion of specific tax streams, and rather than repay asiendos in cash, the king often gave lenders juros as collateral, which they would sell in lieu of cash payments from the crown.

The actual numbers are just mild melting; with fully 60% of the crown's revenue being spent on war, and much of the remainder on Philip's pharaonic palace-monastery El Escorial, the crown's outstanding debt grew by some 20.7 million ducats, and the king had also entered into short term contracts for 83.2 million more between 1560 and 1600. "In an average year, he contracted asientos for 2.5 million ducats, carried total debts of 34.9 million, and had revenues of 6.6 million."

Despite the numbers, it does not seem as though the crown's treasury was a house of cards in this period; Philip II's Spain meets several measures of sustainability, and the defaults occurred in years where revenue was especially low compared to volatile expenses. Military spending was a serious financial gamble; victory could be as lucrative as defeat was ruinous to the crown's finances, and had campaigns in the Netherlands or against England succeeded, it's possible that the king would have been able to meet his debts.

Drelichman has authored a few articles on the Spanish debt crises of the 16th century, one of which you can read here: http://eml.berkeley.edu/~webfac/cromer/e211_f07/voth.pdf

See also his awesomely titled article, 'Lending to the Borrower from Hell'

Geoffrey Parker has written extensively on Philip II, Spain, and war in the Early Modern Period; look up The Grand Strategy of Philip II, The Army of Flanders and the Spanish Road, and Military Revolution: Military Innovation and the Rise of the West, 1500-1800, especially Chapter 2, Supplying War.

Not a historian, but an economist here.

It all depends on how you define bankrupt. If you would say that a country is bankrupt when its state defaults(i.e. does not honor) on its payments to outsiders. Then yes many countries have gone bankrupt. This paper by Reinhart and Rogoff details historical data of state defaults going back to the 14th century.

Their thesis is that defaults are more norm than exception when it comes to the state defaulting (in one way or another) on its obligations. The vast majority of countries (including the US, according to their definition) have defaulted on their payments at some point in time. Some states even have defaulted several times in short periods of times.

As to why states go bankrupt: their government decides for some reason or another that not paying its creditors is the best decision in its current circumstance. The process that leads to this decision (i.e. what ruins the finances of the state) is an active topic in economic research. Reinhart and Rogoff in their notoriously disputed paper argue that when external debt exceeds a threshold of 90% GDP states usually end up defaulting. Countries also default after their economy and state finances have been ruined by wars (Europe had a quite a few after major wars). Finally simply economic and financial crises can cause defaults of several states at the same time. The 1930's depression led to the bankruptcy of Austria, Germany, Greece, Hungary and many other states.

If you want to learn more about this topic i suggest this paper (no paywall!)

Reinhart, C. M., & Rogoff, K. S. (2008). This time is different: A panoramic view of eight centuries of financial crises (No. w13882). National Bureau of Economic Research.

Or you could buy the book they wrote on the base of this research: Reinhart, C. M., & Rogoff, K. S. (2009). This time is different. Eight Centuries of Financial Folly, Princeton University, Princeton and Oxford.