The major airlines always seem to bounce back from Chapter 11 bankruptcy, or another airline steps in to acquire the bankrupt airline's routes and other assets. Continental has been through Chapter 11 twice. United has finally emerged from a particularly long bankruptcy. US Airways, having filed Chapter 11 twice in recent years, was acquired by America West (itself a bankruptcy survivor); the combined airline will retain the US Airways name. A few years ago, American absorbed TWA's routes and many TWA aircraft.
Now, Delta and Northwest are under Chapter 11 bankruptcy protection. I expect both to provide continuous operations through bankruptcy protection, and both to emerge successfully.
Hawaiian Airlines emerged from Chapter 11 protection in June 2005. Aloha Airlines emerged from Chapter 11 protection last month. And ATA Airlines emerged from Chapter 11 protection last week.
On the other hand, there have been other cases where smaller airlines went into Chapter 11 bankruptcy, and subsequently folded; there have even been cases when they went straight into Chapter 7 liquidation. Before buying tickets from any small airline, especially for flights that are many months from now, I would check the financial health of that airline.