However, reports suggest customers are still affected by the IT snafu, with many taking to social media to vent their anger.
Bill Curtis, chief scientist at software intelligence firm CAST, argued that many banks haven't upgraded their IT systems because of their complexity.
The bank said these related to nominee accounts, where certain individuals such as parents or children are able to access another person's account in branch or on the phone.
Almost two million United Kingdom banking customers are reportedly experiencing difficulties using their account online, with some able to access other users' funds after an IT upgrade went wrong.
Other angry customers took to Twitter to express their frustration at not being able to withdraw their money with many even demanding compensation.
"This isn't the level of service that we pride ourselves on providing, and isn't what our customers have come to expect from TSB, and for that I'm truly sorry", said Pester.
Other customers have been unable to log on, others have reported problems with passwords, and some have said that other people's funds have been transferred into their own bank accounts.
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Conor Murphy, CEO at Capricorn Financial said all lenders get service issues from time to time and the net issue is that service to the consumer suffers.
"It's a very concerning development for any customer of any bank to be anxious about where their money has gone", Irish Finance Minister Paschal Donohoe told reporters earlier on Tuesday, adding that the central bank was reviewing the matter.
The bank apologised after customers complained of problems with mobile and online accounts several hours after a system upgrade was supposed to have been completed on Sunday.
The issues are understood to have started when TSB attempted to switch over from an IT system it had been renting from Lloyds Banking Group to a new one. "Potentially millions of customers could be affected by uncertainty and disruption", she said.
A TSB spokesperson said: "We are now experiencing large volumes of customers accessing our mobile app and internet banking, which is leading to some intermittent issues with people accessing our services". I closed the app and went in again at 19:00 and only my account was showing - but my balance was showing at zero.
The Information Commissioner's Office (ICO) and the Financial Conduct Authority (FCA) are assessing the IT meltdown at TSB that led to some customer accounts being seen by other customers. Want to speak to someone now and a half an hour wait! Might just inform them a bank is giving away other people's account numbers.