There have been claims that a world financial crisis is possible if the U.S. government has any delay paying interest or principle on bonds. While I have my doubts about that, there is no reason to expect a world financial crisis if the government fails to pay for goods it already has received much less fail to spend money on what Congress has appropriated. The claim that failure to increase the debt limit creates a threat of world financial crisis is again, dishonest. What is really happening is that the Obama administration is threatening to cause a world financial crisis by failing to pay interest and principle on the national debt and instead spend that money on other things.
Now, suppose the Obama Administration carries out this threat. The debt limit is reached and the treasury just spends appropriated funds as it receives requests for funds. When they run out of cash, they just stop making payments. And if some interest and principle claim show up during that period, then they are not paid.
Some claim that the interest rates the government would have to pay would rise. This seems plausible enough. But the next step in the argument is wrong. The notion that higher interest rates would slow the economy and lead to recession is false.
By purchasing government bonds that the government is selling, the Federal Reserve can keep interest rates on both government bonds and the economy low. As the government sells new bonds as old bonds are repaid (in an apparent hit or miss fashion,) the yield on the bonds can stay low. The interest rate on these bonds will only need to rise to head off inflation. And while that is a very possible scenario, the problem will not be anything like what happens when the Fed increases interest rates by restricting growth in the quantity of money.
Further, it is a mistake to assume that other borrowers would have to pay higher interest rates. Usually, U.S. government bonds are assumed to be risk free. Other sorts of bonds have higher interest rates because of default risk. If the "risk free" interest rate on government bonds should rise, then all of the other interest rates will rise with them, with the "spreads" representing different levels of risk assumed to be constant. But when the government bonds have higher interest rates because of higher default risk, then this is no longer the correct analysis. What happens instead is that some of those who were holding government bonds as a "perfectly" safe asset will respond to the greater risk on them by selling them and purchasing other securities that were slightly more risky. The interest rates on those securities will fall, making it easier, not harder, or the firms issuing those securities to borrow.
Of course, there is another effect. Those holding short and safe assets like government bonds, would likely respond to the risk of delayed payment by holding money instead. If the quantity of money fails to expand enough, the resulting liquidity squeeze will tend to raise all interest rates. But then, that is why the Federal Reserve would need to expand the quantity of money in such a situation. It can and should supply the extra money that people would want to hold, and so avoid a liquidity crunch.