Ursula Lyons found the card problem "annoying and embarrassing", after experiencing issues at a cash point and accessing her online banking account.
But Tracy Pomphrey said: "Myself and my mum had our TSB cards declined in Asda, it was extremely embarrassing. Although we could go out to the cash machine outside and withdraw the cash!"
Hayley Jane said: "People saying they are glad they're with other banks, but they have all had this problem at some stage - Natwest had this problem last year for ages!!"
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The letter was signed by the CBI, British Chambers of Commerce, the EEF, the Institute of Directors and the Federation of Small Businesses.
A spokesperson said Rory Stewart's remarks were in line with the British government’s stated position.
The minister's intervention will come a week after Trump decertified the landmark agreement claiming it was not in the US national interest.