One woman, Josie Simpson, told the BBC she called the airline 67 times to try to rebook a family holiday to Florida.
The company’s Twitter feed has also been inundated with messages from frustrated people.
In response to one passenger, a BA representative said: “We’re extremely sorry that you’re having difficulties trying to rearrange your flights.
“Our teams have been working tirelessly to help as many of our customers as possible, in these unprecedented circumstances.”
Some customers who were told their flights were not scrapped after all have been left confused about whether their decision to accept a refund has now been cancelled.
Others said they fear being left out of pocket.
Ellie Kormis, from Surrey, spent the whole of Saturday trying to speak to BA after being told one of her flights for her family’s package holiday to Greece was cancelled.
She said they ended up booking new flights – which extended their holiday by three days – and extra accommodation, costing more than £2,000.
“You’re left in a situation where you can’t speak to anyone – and you fear you’ll either lose your holiday or be left out of pocket,” she said.
BA then contacted her by email to say her flight wasn’t cancelled after all.
She said it was “an epic mess up on their part”, joking that she had lost hope she would ever get through to speak to someone at BA.
A BA spokeswoman said customers who were told their flights were cancelled in error and had booked alternative flights or incurred other expenses would be entitled to refunds if they submitted receipts.
Earlier, some people said were told that wasn’t the case.
Kaelee Matthews, from Cardiff, said BA had told her it wouldn’t give her a refund for her flights to Orlando – which the airline had initially said were cancelled – because they were still operating.
She has booked new flights with Virgin for her and her partner – costing an extra £440 in total.
“We are disgusted with BA,” she said. “We don’t know what to do now. Virgin can’t refund us, but we understand that. Travel insurers say we’re not covered either.”
Laura Gillespie, 48, from Perth, was also told by BA she was no longer entitled to a refund as her flight from London to Edinburgh was going ahead after all.
“I’ve now got flights booked with two different airlines going to the same place and I’m £140 down,” she said.
“I know it’s not a lot of money compared to some folk who have spent thousands but it’s so annoying.”
In response to customers being frustrated at not being able to get through to customer services, BA said:
It received 38,000 calls and 33,000 tweets in first 24 hours
Contact centres stayed open 24 hours to help resolve issues, with 70 extra staff
Around 100 staff were working to answer Twitter queries
The British Airline Pilots Association (Balpa) said on Friday the strikes were a “last resort” born out of “enormous frustration” with airline management.
Pilots have rejected a pay increase worth 11.5% over three years, which the airline put forward in July.