A computer failure at TSB that caused up to 1.9 million people to lose access to online banking services is being investigated by the financial regulator.
The introduction of a new IT system in April left customers struggling to make transactions and see their balances.
The Financial Conduct Authority (FCA) said the bank's boss, Paul Pester, had given an "optimistic view" of services.
He is due to appear before MPs for a second time on the crisis later.
FCA chief executive Andrew Bailey said: "We do not normally make this information [the investigation] public, but, given the level of public interest, I want to be clear that we will be conducting this work." The FCA has powers to impose unlimited fines.
He added that the bank's poor communication may have hit customers' trust in banking.
Some TSB customers are still facing problems with services after its five million customers were moved to a system run by its Spanish owner Sabadell six weeks ago.
In the midst of the crisis, Mr Pester told MPs on the Treasury Committee that most TSB customers could carry out transactions as normal.
In a letter to the committee, the FCA's Mr Bailey said that view was optimistic and too positive.
"Greater caution would have made sense," he wrote, adding that Mr Pester should have shared more details about the problems with MPs on the committee.
The customers themselves were told by the bank that the vast majority of them could use their online accounts, when only half could do so.
Mr Bailey wrote: "The FCA has been dissatisfied with TSB's communications with its customers and we have had concerns that TSB was not being open and transparent about the issues experienced."
One TSB customer told BBC Radio 4's You and Yours last month that he watched thousands of pounds in wedding savings being stolen from his internet account as he waited on hold for the bank's fraud department.
Ben Alford, from Weymouth in Dorset, said it took more than four hours to get through to TSB, by which time most of the money had gone. TSB said it had put in "additional resources" to support customers.
Lorna Connolly, formerly Lorna McHale, told the BBC how she was unable to access her account days before her wedding day.
The 26-year-old said she had to "ring to grovel" with suppliers for the wedding. TSB rang the day before her wedding to offer her compensation of £100.
Richard Lloyd, chairman of the disputes resolution website Resolver, told the BBC's Today programme that his organisation was still getting about one complaint an hour from disgruntled TSB customers.
He called on the FCA to make the compensation rules much clearer because TSB's "communication with customers has been atrocious".
Nor was TSB alone, he added. Banking IT systems were "creaking", and the TSB failure "doesn't seem to be an "isolated incident".
What caused the problems?
When TSB split from the Lloyds Banking Group, it continued to use its computer system while a new one was developed.
When it was ready, TSB moved customers' data from the Lloyds platform to its own.
This was a long-planned disruption to the service. The bank said it informed customers of the change, and that it would lead to them being unable to use online banking or payment systems that weekend.
That led to two problems. First, many customers said they were unaware of the changes and so were caught out. Second, customers experienced difficulties long after the deadline that TSB had promised things would be fixed.
Many services have improved, but the situation remains unstable, with 40% of those trying to call the bank unable to speak to someone, while waiting times have run to more than 30 minutes.
Fraud has become a problem, with confused customers being tricked into allowing access to their accounts. Yet the FCA said that TSB had failed to refund their money quickly enough.
The FCA has had its own team embedded with TSB since the crisis unfolded, with as many as 14 supervisors from the regulator and the Prudential Regulation Authority (PRA) deployed.
This will be the FCA's first major investigation into an IT meltdown since RBS was fined £56m by regulators after a 2012 software issue left millions of customers unable to access accounts. That investigation, carried out jointly with the PRA, took 18 months.
Nicky Morgan, who chairs the Treasury Committee, said she was "deeply concerned by TSB's poor communications about the scale and nature of the problems it has faced, by its response to customer fraud, and by the quality and accuracy of the oral and written evidence provided by Mr Pester to the committee".
The committee will question senior members of the FCA and TSB bosses on Wednesday afternoon.
A TSB spokesperson said it was "doing whatever it takes to put things right for our customers and ensuring that no customer will be left out of pocket as a result of the recent IT issues".