What happens to a country with debt depends on several factors.
- Debt to GDP ratio
Debt is not exactly free money. It is 'free' money (apart from interest) only if you can be depended on to pay back that debt. In other words, investors are willing to lend money to a government if they are sure that the government will pay them back. Failing to pay back your debt would require your country to declare bankruptcy (or drop out of the global financial system altogether).
The reason why governments like the U.S. can keep borrowing money is because investors believe that the U.S. can easily make enough money to pay them back in the future. This is based on the debt to GDP ratio. If your debt to GDP ratio is too high (say, 10 to 1) that means it might be quite difficult for the country to pay back all that debt. The current U.S. debt to GDP ratio is between 0.7 and 1.0, which is relatively high (a result of the Great Recession).
Another smaller country might not be able to easily borrow with such a high debt to GDP ratio, but the U.S. also has credibility - it has not defaulted on its debts before, unlike some other sovereign states (Argentina). Combined with the fact that the U.S. has large financial clout and is relatively important, investors feel that the U.S. government is unlikely to default on its debts.
If the U.S. government were to keep on borrowing money, eventually people would get a little concerned with the amount of debt it has taken on.
It's possible to think of sovereign debt in a similar way to personal (your) debt. For example, if you own a credit card, you can use that credit card to get 'free' money and use that money to purchase things. The reason why banks are willing to lend you this money is because you have previously paid them back. If you were to (1) ask for a lot more money than you make (e.x. $100 million) or (2) ask for money after failing to pay the bank back at the end of the month, the bank would reject you. Investors, like the bank, require governments (and individuals like you) to meet some requirements before they are willing to lend you money.