The media popularly equate 'bankruptcy' and 'Chapter 11' together. This is because the US Bankruptcy Code is a body of legislation which is divided into a number of Chapters, which have different applications for different situations. Chapter 11 of this code is designed to assist companies in financial difficulties from avoiding payment of most of their immediate bills and obligations in order to give the business time to reorganize itself and, ideally, emerge from Chapter 11 down the line in a stronger position. A business requires court permission to enter Chapter 11.
Thus, paradoxically, when a company files under Chapter 11, it is ironically NOT bankrupt. It is, in fact, technically called bankruptcy protection, or protection from creditors. It is a still a live and active business, merely under new temporary management. In the case of the airlines, passengers rarely notice any difference.
Under Chapter 7 of the Code you would have bankruptcy, as filing under Chapter 7 you file for liquidation of the business. If United, for example, didn't come out of Chapter 11 (it plans to late this year, in fact), it would liquidate under Chapter 7, and only then would become bankrupt.
The question of whether the airlines in the US are abusing Chapter 11 is arguable and a very live issue right now. On balance, they probably are. I am sympathetic to the poster who says we need one big carrier to go out of business for shock value purposes, if nothing else.