"I take full personal responsibility for our being in this situation," Arpey said.
Still, travelers like Jeff Ostrove fill the airports. Ostrove was stuck Thursday morning in Los Angeles on the latest leg of his tortured journey from San Diego to Tortola in the British Virgin Islands.
"We had a flight from San Diego through Dallas-Fort Worth to San Juan [Puerto Rico] down to Tortola, and they called us yesterday morning and told us it was canceled," Ostrove explained.
He was rebooked through Los Angeles, Chicago and San Juan, but then received an e-mail informing him of another cancellation.
"Basically in the middle of the night, 2:30 in the morning, we left San Diego and drove up here, and they are getting us through Miami to San Juan to Tortola," he said.
This week's cancellations came in addition to hundreds of cancellationsin late March. The Federal Aviation Administration called for inspections among all carriers after slapping Southwest Airlines with a $10.2 million fine for failing to adhere to requirements for safety and inspection checks.
All airlines are now beginning phase two of the inspections called for by the FAA, which are slated to be far more extensive than phase one.
Arpey said the FAA maintenance directive currently in question for American's wire bundle inspections is 38 pages long.
Still, cancellations at American on their own are making a tremendous impact. Data released Thursday by the Bureau of Transportation Statistics revealed that in January of this year, American Airlines carried 7.7 million passengers — more international and domestic passengers combined than any other U.S. airline.
"As passengers, we should be glad that the FAA is ensuring that the planes we fly on are safe," Jim Hall, former chairman of the National Transportation Safety Board, said Thursday. "I think the FAA is trying to make a statement, and I think that the message will hopefully be heard by all the airlines, and we won't have to see this repeated by each major carrier in the United States."
ABC News' Zach Wolf contributed to this report.