Paul Pester, chief executive of TSB, received 1,300 tweets from customers who could not access their accounts

A TSB adviser warned the bank against embarking on a major internet upgrade on the cheap, The Times can reveal.

Consultants raised concerns in 2015 that the bank was not setting aside enough money for the “incredibly complex” project to move customers to a new online system.

One said: “There have certainly been some examples of where it has cost significantly more to re-platform an entire bank. It is not overly generous as a budget for that scale of migration.”

TSB customers were still struggling to access accounts yesterday despite Paul Pester, the chief executive, claiming that everything was working.

Mr Pester wrote on Twitter: “Our mobile banking app and online banking are now up and running. Thank you for your patience.”

By last night,