MoneySavingExpert.com warned its findings indicate that several high street giants still need to up their game as new rules are about to come in which will make it much easier for their customers to ditch their existing bank.
Less than half of current account customers with Barclays, NatWest/RBS, Lloyds TSB, HSBC and Bank of Scotland said they got a great service from their bank.
The consumer help website, which compiled its research from the opinions of more than 8,000 current account customers, also found that Santander and Halifax have made some strong improvements in recent months.
The findings come just weeks before new rules to take the hassle out of switching current accounts come into force, which are expected to inject more competition into the market.
Providers often use the relationship they have with their current account customer to sell them other products such as mortgages or credit cards. From September 16, the time it takes to switch a current account will be cut from up to 30 working days to seven and incoming and outgoing payments will automatically move to the new account.
First Direct came out on top in a similar study run by consumer group Which? in July. Earlier this month, it upped an incentive for people to switch to it from £100 to £125.
At the other end of the spectrum, more than one fifth (21%) of Bank of Scotland current account customers rated the service provided by the bank as "poor", while just over a third (36%) said they got a great service. The bank was rated as "OK" by 43% of customers.
A Bank of Scotland spokesman said that getting customer service right is at the "heart" of its strategy and the level of banking complaints it is seeing has been falling.
Santander, which was languishing at bottom of the table one year ago, has moved into fifth place for its accounts generally and was also in third place for its flagship 123 account, which offers cashback on household bills paid through the account as well as in-credit interest.
Santander recently announced it is simplifying its current account range, including moving customers off its older "packaged" paid-for current accounts, which it stopped selling more than a year ago.
Earn £100 or up to 5% interest by switching to another bank
Halifax also saw a significant jump in the proportion of customers describing the provider as "great" - up from 38% six months ago to 53%.
Halifax's strategy in preparation for the new switching rules has been based around trying to make sure customers are kept happy once they have changed to it.
It is offering customers £100 to switch, as well as a fee-free overdraft for the first six months. Halifax has also recently cut the amount of cash people need to place in its Reward and Ultimate Reward current accounts in order to be paid £5 into their account each month from £1,000 to £750 a month.
Dan Plant, head of editorial at MoneySavingExpert, said the findings show that several major banks need to "step up their game" as the new switching service is likely to bring lots of offers from providers trying to tempt new customers in.
He added: "We are now just weeks away from the new switching rules so there really is no excuse to put up with bad accounts with even worse service."
:: Here is how the current account providers were ranked in terms of customer service. The name of the provider is followed by the percentage of customers who thought service was "great", followed by those who thought it was "OK" and lastly, those who thought it was "poor". The figures in brackets show how the providers performed when the same study was carried out in February. The figures have been rounded:
1. First Direct, 93% (91%), 6% (7%), 1% (2%)
2. The Co-operative Bank (including Smile), 76% (78%), 18% (18%), 6% (4%)
3. Santander (123 account only), 75% (63%), 20% (28%), 5% (9%)
4. Nationwide, 73% (66%), 22% (28%), 5% (6%)
5. Santander (all accounts), 62% (44%), 27% (37%), 11% (19%)
6. Halifax (Reward account only), 56% (n/a), 35% (n/a), 9% (n/a)
7. Halifax (all accounts), 53% (38%), 37% (43%), 10% (19%)
8. Yorkshire/Clydesdale Banks, 51% (n/a), 40% (n/a), 10% (n/a)
9. Barclays Bank, 48% (41%), 40% (45%), 13% (14%)
10. NatWest/RBS, 48% (40%), 39% (44%), 14% (16%)
11. Lloyds TSB, 47% (44%), 40% (42%), 13% (14%)
12. HSBC, 44% (42%), 42% (44%), 15% (14%)
13. Bank of Scotland, 36% (35%), 43% (43%), 21% (22%)