See http://www.telegraph.co.uk/finance/economics/8639896/The-las... for how a similar situation in 1979 caused us to accidentally pay $120 million slightly late, resulting in our paying several billion in additional interest.
Could the federal reserve theoretically give a forbearance on interest payments? The debt owed to the public and international financiers (China,Japan) is fairly insignificant compared to what's owed to the Federal Reserve, and could probably be paid.
What happens if we pay everyone except ourselves?
We are headed towards a situation where we're in default, no matter what the Constitution says. If that happens our creditors will be in a position to go to the courts and get their money back. It will be a slam-dunk. They will win. But on the date that they are supposed to have their money, they still won't have money to spend. And that is a big problem.
Incidentally the actual text of the 14th amendment doesn't say anything about "default". It says, The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. In other words if you've been told that we owe you money, we owe you money. Period. But it doesn't actually say when or how you'll get that money.
Incidentally if we do default on some of it, we should, unlike what we did in 1979, pay it back with a generous penalty on top. That would do a lot to make lenders more comfortable that it isn't so bad to temporarily get paid late. For instance in 1979 the pre-emptive spending of, say, $40 million would have saved us billions over the next few years.
Unfortunately nobody has the political will to consider that course of action, no matter how good it would be...
The problem is that the Tea party people can't afford to compromise their position at all and even many who are not members of it are still under their influence.