The first problem left Nationwide customers locked out of internet banking and mobile apps.
The society said the IT problems started yesterday morning and were "intermittent", with some users who could not log on finding they were then able to do so a couple of minutes later.
Nationwide, which is Britain's biggest building society, said not all customers were affected by the problems, which its IT team was working to fix.
It said the latest issue is unrelated to a separate technical glitch which occurred on Wednesday.
Meanwhile, Barclays customers vented their frustrations on Twitter after being left unable to use the company's online banking services or withdraw cash from cashpoints.
It was due to a temporary fault on the system lasting 20 minutes, starting at around 3pm yesterday.
Author India Knight tweeted: "Is something the matter with Barclays' system? Post Office says my debit card 4th to be declined." She later added: "Maybe have a Plan B if you're planning to use your card/a Barclays ATM."
The bank used the social networking site to say it was dealing with the issue.
Its statement said: "We're aware of an issue that is currently affecting some customers when attempting to make card transactions and withdrawals."
"We are working to resolve this issue as quickly as possible and apologise for any inconvenience."