TSB says its internet banking service is currently working at around 50% capacity for customers as chief executive Paul Pester said he was "truly sorry" for the ongoing problems.
The bank said it was working "round the clock" to fix problems that struck following a major IT upgrade on Sunday night, but said five out of every 10 customers who tried to access its internet banking service were still facing issues.
Its mobile app was now operating at 90% capacity, it said.
Mr Pester said: "Our teams continue to work around the clock to fix the problems that some of our customers are having in accessing their TSB accounts.
"I want to reassure our customers that the engine room of the bank is working as it should. This means that for the vast majority of our five million customers, everything is running smoothly.
"The challenge we are facing at the moment is that while we know everything is working, one of the main ways that our customers see everything is working - through our internet banking and mobile app - isn't functioning as well as it should be, and for this I'm truly sorry. I can appreciate how frustrating this must be for our customers.
"At this time, our call centres and branches may be busier than usual helping our customers.
"While we will try and help everyone as quickly as we can, there are a number of other ways to bank with TSB.
"For instance, customers can check their balance and withdraw money at a cash machine and they can also, as always, do this and make payments through their local Post Office.
"Of course, customers can rest assured that no-one will be left out of pocket as a result of these access issues."
Mr Pester announced on Twitter on Wednesday morning that the services had been restored and thanked customers, many of whom had been left furious by the outage, for their "patience and bearing with us".
But a spokeswoman said the bank was limiting the number of customers to ensure those able to access its app and online banking received a normal service.
TSB had intended to complete a major IT system upgrade on Sunday evening, although problems persisted into a third day.
Delays meant customers were left unable to access their money online, while some reported being able to access other customers' accounts.
Data and banking watchdogs confirmed they were looking into the situation while MPs have demanded answers from the bank about the IT meltdown.
Treasury Select Committee chairwoman Nicky Morgan wrote to Mr Pester asking for details of the problems and the bank's response.
She said: "This is yet another addition to the litany of failures of banking IT systems.
"Potentially millions of customers could be affected by uncertainty and disruption.
"It simply isn't good enough to expose customers to IT failures, including delays in paying bills and an inability to access their own money.
"Warm words and platitudes will not suffice. TSB customers deserve to know what has happened, when normal services will resume, and how they can expect to be compensated."
Customers complained of problems with mobile and online accounts several hours after the system upgrade was supposed to have been completed.
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.
A number of customers complained they were still unable to access their money on Tuesday.
The Information Commissioner's Office, which monitors data and privacy, said it was "aware of a potential data breach in relation to TSB and are making inquiries".
Banking regulator the Financial Conduct Authority said: "We are aware of the issue and are liaising with the firm."