But it was United Parcel Service, the world’s largest package delivery company, that was perhaps the most unprepared for the crush. The company hired 55,000 seasonal workers this year, but that number was roughly the same as last year and the year before that — not enough to keep up with rising demand.

“It hasn’t fluctuated that much over the past couple of years,” Natalie Black, a spokeswoman for the company, said of its holiday staffing. “Whether that was part of the problem, I can’t say. Right now, we don’t know what the linchpin was for the network breakdown.

“You can only fit so much in planes,” she added.

It was unclear how many customers were affected, but complaints poured in from across the country and retailers large and small were caught up in the maelstrom.

While bad weather and a short holiday shopping season were cited as possible causes by U.P.S. officials, they also said the volume generated by growth in online shopping was a likely factor. Online sales have been growing for years, and this season, the rise during the weekend before Christmas was extremely steep, up 37 percent, according to IBM Digital Analytics Benchmark. FedEx said that it had predicted it would deliver 22 million packages on its busiest day this year — double the volume in 2007. The actual number is not yet known. One way to address future demand, Ms. Mulpuru, the analyst, suggested, would be to approach the surge the same way that the airlines do: by charging more for the service.

“An airline doesn’t just buy additional aircraft so they can accommodate everyone who wants to fly the day before Thanksgiving for $300,” she said. “They just raise the price of your ticket and force people to go earlier.”