TSB was apologised after a weekend systems upgrade left customers experiencing serious problems with their online banking.
In one case a customer reportedly discovered they had been credited with £13,000 after logging back into the system.
- What is going on?
TSB has been migrating its online system away from former owner Lloyds Bank.
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.
The upgrade window was scheduled between Friday at 4pm and Sunday at 6pm.
But some customers complained they were still unable to access their money on Monday morning.
- What are customers saying?
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.
"I suggest somebody answers the phones as iv been on hold for 45 minuets!
"This is a MASSIVE breach of data protection! If i have access to their account they could have to mine as well!"
Another Twitter user called Bex said:
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.
- What hasTSB said?
The bank has confirmed there are "intermittent issues" with its mobile and internet banking services.
A TSB spokesperson said: "We are currently experiencing large volumes of customers accessing our mobile app and internet banking which is leading to some intermittent issues with people accessing our services.
"We are really sorry for the inconvenience this is causing our customers and want them to know we are working as hard and as fast as we can to resolve this problem."
- Why is the migration happening?
TSB was bought by Catalonia’s Banco Sabadell in 2015, but the online services migration has taken some serious time - the original target was November last year.
The bank says using the Lloyds platform is expensive. It has previously said that using its own platform would save it £100m a year.