Lloyds Banking Group said the faults were not caused by any external upgrade work or cyber attack.
Many customers found their debit cards were declined and about 3,500 ATMs were not working for three hours.
The technical fault has now been fixed, and the banking group has apologised to those affected.
The hardware failure came on a normal day for the banking group, it said.
"We were in 'business as usual' mode - and there was no maintenance or update activity on the system at the time," said a spokeswoman for Lloyds.
By Kevin PeacheyPersonal finance reporter, BBC News
This is not the first bank to suffer from a computer failure, and it certainly won't be the last.
New technology will put even more strain on the banks' creaking IT systems, which were built to run a very different banking system decades ago.
So how do we cope as consumers?
The irony is that the more we embrace the digital wallet, the more emergency cash we may need to carry for when these systems crash.
"We are working with our suppliers to understand what caused the hardware failure, but there is no evidence that it was as a result of an external intervention or attack on our systems.
"We continually review and update our systems as necessary and make significant investment to do this."
The chief executive of TSB, Paul Pester, said on Sunday that its customers' debit card transactions had been affected after problems with two out of seven IT servers.
He tweeted an apology and replied to some comments personally.
Lloyds Banking Group serves about 30 million customers and is the largest retail banking group in the UK. TSB has about five million UK customers and is in the process of being split from Lloyds Banking Group.
The Financial Conduct Authority (FCA), the City watchdog, said it had been in contact with Lloyds and would be updated on how customers had been affected.
Not everyone who tried to use their cards suffered problems, and Lloyds said that online customers were unaffected.
UK banks have suffered a series of IT problems over the past few years. In December, an estimated 750,000 customers at the Royal Bank of Scotland group were unable to use their credit and debit cards for several hours.
Leanne Seaward, 29, from Verwood in Dorset, said she discovered she had problems when she went to pay for her weekly supermarket shop at about 16.15 GMT on Sunday.
"It was a little embarrassing, I put my card in and it kept saying transaction void. I thought it was because I am in the process of switching banks, so assumed they might have closed my account without telling me.
"Luckily I had my husband with me so he was able to pay, but if I was getting petrol and on my own it could have been a completely different matter."
Your experiences
A week later, NatWest - which is part of the group - said that it was hit by a cyber attack. A deliberate attempt to disrupt services by creating a surge in traffic - known as a distributed denial of service attack - meant some customers struggled to access online banking.
In 2012, a major IT failure locked many RBS, NatWest and Ulster Bank customers out of their accounts for several days. The issue was particularly bad for Ulster Bank, with customers facing access problems for weeks.
Most banks have seen online and card failures in recent years, with many of the computer systems being used dating back decades.
TechUK, a trade body for a number of technology companies, said: "It is now widely acknowledged that the technology infrastructure across many financial institutions is exceptionally complex, to the point where it no longer serves many banks, it hinders them.
"However many financial institutions are unable to address these issues because of the sheer cost and disruption that replacing these systems would entail."
A hardware failure was the cause of the shutdown of some Halifax, Lloyds, Bank of Scotland and TSB cash machines and debit cards on Sunday.