Finance and data watchdogs are looking into IT issues at TSB that left online customers unable to access their money and some able to see other people's accounts.
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.
One customer said they had "access" to other customers' accounts totalling more than £20,000 and another reportedly discovered they had been credited with £13,000 after logging back in.
READ MORE: Can regular saving with TSB’s Loqbox improve your credit rating?
A number of customers complained that they were still unable to access their money on Monday morning.
The Information Commissioner's Office, which monitors data and privacy, said it was "making inquiries", the Financial Times said.
Meanwhile banking regulator the Financial Conduct Authority told the paper it was "liaising with the firm".
TSB said a "tiny fraction" of customers saw accounts belonging to other customers and the issue was resolved in about 20 minutes.
The bank had told its account holders that some of its services, including online banking, making payments or transferring money, would not be possible over the weekend.
READ MORE: Can regular saving with TSB’s Loqbox improve your credit rating?
The upgrade window was scheduled between Friday at 4pm and Sunday at 6pm.
However a message on the TSB website on Monday morning said there were still "intermittent issues" with its mobile and internet banking services.
The bank said it was "working as hard and as fast as we can to get these up and running".
TSB's social media team faced a stream of complaints from customers, including several who said they could see other people's accounts after logging back in on Sunday evening.
Craig Malcom tweeted: "@TSB I currently have access to £20k+ of other peoples money.
"This is a MASSIVE breach of data protection! If i have access to their account they could have to mine as well!"
READ MORE: Can regular saving with TSB’s Loqbox improve your credit rating?
Another Twitter user called Bex said: "@TSB so go to my app and have someone else's accounts there!!!! Serious dpa (Data Protection Act) breach! Want to speak to someone now and a half an hour wait!! What would the FCA (Financial Conduct Authority) say about this!!! Might just inform them a bank is giving away other people's account numbers".
Laim McKenzie, from Paisley in Scotland, told the BBC he had been unexpectedly credited with thousands of pounds after logging back in after 6pm on Sunday.
"My balance, because of my overdraft, is in minus, but my balance was showing at £13,000," he said.
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