As things stand, the US Treasury is going to run out of money next Tuesday, and the USA will no longer be able to pay its bills. There is a cap - currently 14.294 trillion dollars - on the amount of debt the Treasury can issue, which is preventing it from borrowing any more money. It needs agreement in Congress to raise the limit, and that agreement has not been forthcoming, because some Republicans - enough to block progress in the House of Representatives - are unwilling to agree unless they get their own way on absolutely everything. Increases of the limit usually happen about once a year, and approval has been used before for political bargaining, but never to anything like this extent.
There's been some discussion of the notion that if the cap is not raised, since the Executive is going to have to ignore some of its instructions from Congress, it might as well ignore the cap. Obama says his legal advice is that he can't do that, but speculation continues...
I think it more likely than not that some sort of a deal will be reached just in time. If it is not, I would expect the Federal Reserve to find a way to "print" electronic money to keep things going. And if that fails, they will prioritize debt payments, so the US will not default on its bonds (though it would default on its obligations to employees and pensioners).
Nevertheless, the bond markets are not going to take this lightly. Bond investors care very much whether they are going to get paid back or not (and credit ratings are no more than a guide to that, so the possibility of a ratings downgrade is not the whole story). The fact that powerful politicians are willing to play chicken with the USA's ability to pay its debts will already have affected the market's confidence.
As it happens, US Treasury bond prices have not fallen. Investors might be somewhat shaken by what's going on, but they can't think where else to put the money - the global financial system would be rocked by any sort of a US default, and US Treasuries would probably remain a relatively secure investment, especially compared with anything else denominated in dollars. But that could change. At some time in the future, the US is going to find the markets demanding higher yields because it knows that US politicians don't take paying their debts that seriously.