The parents, who did not disclose their last names for fear of repercussions, and their infant, identified only as Riyanna, had boarded JetBlue Flight 510 from Fort Lauderdale, Florida, to Newark, New Jersey, on Tuesday evening when they say an airline employee asked them to get off the plane.
"I said, 'For what?'" Riyanna's mother said. "He said, 'It is not you or your husband. Your daughter was flagged as no fly.'"
The family is of Middle Eastern descent and the mother wears a headscarf, but they say they are US citizens and have lived in New Jersey all of their lives.
However, the government disputed the airline's initial characterisation.
"Transportation Security Administration (TSA) did not flag this child as being on the no-fly list," CNN quoted Transportation Security Administration spokeswoman Sterling Payne as saying in a statement.
"TSA was called to the gate by the airline and after talking to the parents and confirming through our vetting system, TSA determined the airline had mistakenly indicated the child was on a government watch list."
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The family declined to continue on the flight, saying they felt humiliated.
"We were put on display like a circus act because my wife wears a hijab," Riyanna's father said.
The airline says it was a technical mistake.
"We are investigating this particular incident. We believe this was a computer glitch," JetBlue Spokeswoman Allison Steinberg said in a statement without elaborating.
"Our crew members followed the appropriate protocols, and we apologise to the family involved in this unfortunate circumstance."
"The whole situation was bizarre, it was completely bizarre and absolutely made no sense," the mother added.
Steinberg said JetBlue crew members "are trained to address each situation discreetly, treating every customer with dignity and respect."
The government maintains several databases of individuals who warrant extra screening when they fly. No one on the official no-fly list would be given a boarding pass, according to a TSA official who did not want to be named discussing the matter.
About 20,000 people are on the no-fly list, according to a counter-terrorism official, including 500 Americans.