RBS and Natwest customers experienced problems accessing their bank accounts online and via the telephone at the weekend following an IT systems upgrade.
It came just a day after HSBC's online banking systems went down as a result of a mainframe problem.
An RBS spokesperson said: "We experienced systems problems throughout the day on Saturday, which affected our customers and apologise for any inconvenience caused.
"The issues emerged as a result of major system changes that took place overnight on Friday. We are disappointed that we were not able to fully resolve all of the issues in the timescales that we wanted."
Problems with the systems upgrade meant that account balances were not updated overnight, and some customers were unable to withdraw cash from ATMs.
The bank said that during the problems on Saturday, its ATM, mobile and credit and debit card payment systems were operational.
Its online banking and telephony "are now running largely as normal" and account balances have been updated, it said today.
"Additional resources were mobilised, including extra staff in our branches on Saturday with a number of our branches extending their opening hours.
"Customers who faced inconvenience as a result of these problems should contact us in the usual way. We will refund all charges applied as a result of this weekend's problems," the bank said.
RBS's situation is one of a string of problems to hit banks' online systems recently.
On Friday afternoon, HSBC's customers were unable to use their online banking services, bank cards and ATMs after a "mainframe outage".
In an update on the situation at 6.30pm, the bank said that all systems are now working, but those with any issues should contact HSBC on 08457 404 404.
Meanwhile, some Bank of Scotland and Halifax customers are still experiencing problems with accessing their accounts online nearly two months after parent company Lloyds Banking Group carried out a systems migration.