I didn't find out about Braniff's demise until about 10:00 that night, but took a chance and called the travel agency who had done the ticketing. To my surprise, someone actually answered. Evidently after the news came out, the travel agency called all of it's staff back in to man the phones. I remember they rebooked my friend on AA with no problem.
Also, what if you purchase the tickets with a credit card? If you don't get the service you paid for, I believe you could dispute it with the credit card company and get your money back and the credit card company would have to deal with recovering the funds from the airline, but I'm not sure if this is the case.