In Southwest’s case, a backup system was in place, but the airline said that system was not triggered as it should have been when the router failed. And Delta said on Monday that it was investigating why some of its own critical operations had not switched over to backup systems.

“In the case of Delta, whatever occurred was clearly a catastrophic failure, and it is alarming that the backup system didn’t kick in,” said Henry Harteveldt, a travel industry analyst at Atmosphere Research Group.

Delta said the problem was touched off by a power failure about 2:30 a.m. Eastern, shutting down computers and grounding flights for about six hours before the airline began to bring its systems back online. (As it happens, the culprit inside Delta — a failed switchgear, an item similar to a circuit-breaker box in a house — is a piece of equipment typically installed to guard against breakdowns.)

Throughout the day, the consequences were playing out in human terms.

In Terminal C of La Guardia Airport, a central hub for Delta in the New York City region, it was standing room only Monday morning, with children splayed out napping on backpacks, business travelers scrambling for outlets, and many passengers leaning against the cylindrical pillars for support.

As the morning wore on and the system began to slowly reboot, occasional problems also caused more confusion.

“For those of you traveling to Detroit on Flight 831,” a gate agent said over the loudspeaker, “you may have just received a text message saying we are departing at 9 a.m.” That, she indicated, would be great, but also nearly impossible. The computer system had shut back down before the pilot’s papers, including the flight plan, could even be processed.

In Phoenix, Anthony Navarro, 25, was headed to Atlanta, and then to Miami for a cruise to the Bahamas. He and a friend boarded a flight scheduled to leave Phoenix at 12:35 a.m., but then sat in the plane, parked at the gate, for about five hours.