Earlier this week, the chairman of the OFT, Philip Collins, re-stated the regulator's long-standing view that the banking market was still not working well for current account holders.
He said it needed to be one in which "properly informed consumers manage their accounts and make active choices about which account provider to use, switching where necessary in response to individual and changing needs and new competitive offers".
But the British Bankers' Association (BBA) said the survey showed that the silent majority were happy with their bank's service.
"What the banking industry has done over the last two to three years, in some considerable depth, is asking all its customers, 'What is it that you want?' and has been acting on the replies that it has got," said Angela Knight, chief executive of the BBA.
92%: not changed their bank account in the last two years
93%: happy with service from their banks
48%: thought that switching accounts would cause problems
Source: ICM poll for the BBC
Of the bank account-holders questioned, 896, or 92%, had not changed their accounts in the past two years, despite the huge wave of bad publicity surrounding bank overdraft charges, the mis-selling of payment protection insurance, and the tens of thousands of complaints that the banks attract each year.
In February, the Financial Ombudsman reported that the top five High Street banks accounted for more than half the 82,000 complaints it had received in the second half of 2009.
Last year, the Financial Services Authority revealed that in the 2006-2008 period, more than nine million individual complaints were made directly to financial firms and more than half of them had been about banking and loans.
Yet when asked why they had not changed banks, most of those surveyed for the BBC said they were happy with the service (93%).
Other responses pointed to some of the issues that the authorities have been highlighting.
Half of all those who had stayed put said they thought that all current accounts were the same; 48% felt that switching might create too many problems; 38% agreed that there were no better or more appealing current account deals on offer; and 30% said they did not have time to look around.
"I don't think I'd save with [my bank] or building societies at the moment"
The small number in the survey who had changed account in the past two years (76, or 8%) cited various factors such as better interest rates or overdraft charges, a lack of local branches, problems with customer service, or introductory offers by other banks.
The OFT believes that more people will switch as they study the value for money of accounts.
"As consumers become more aware of the costs of their account and more confident in switching as a means to get better value, so banks will need to offer more competitive and innovative products and services to attract as well as retain customers," said John Fingleton, OFT chief executive.
The stability of banks in the recession was mentioned by just 36% of this already small sample.
When asked about the vexed question of overdraft charges, which saw the banks win a convincing Supreme Court ruling late last year, 80% of all those in the survey thought that banks should only levy charges on people who went overdrawn without permission.
By contrast, 18% felt that banks should charge all customers a fee for having a standard current account.
With recent evidence of a continued rise in fraud attacks on online bank accounts, 60% of all those questioned in the survey felt online banking was secure, but 33% said this way of handling their money was insecure.
"The numbers show that we are getting big increases in people using online banking, but we have not seen anything like the same increase in fraud," said Mrs Knight.
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