Northern Ireland MPs raised the issue of the technical problems which hit Ulster Bank customers during Prime Minister's Questions, as the financial institution says it could be another two weeks before normal service resumes.
Customers throughout Northern Ireland - and also in the Republic - have not had proper access to their accounts because of a technical glitch which occurred on Tuesday 19 June.
Ulster Bank is understoood to be suspending its voluntary severance programme as it battles through the crisis.
Parent company RBS said it now expects "the vast majority of customers" to be up to date by the week starting 16 July.
"We now expect that next week (commencing 9 July) will be the final week of any significant delays for Ulster Bank customers," RBS said.
"We expect gradual, but significant and noticeable improvements throughout the remainder of this week and next.
"It is our expectation that by the week of the 16 July the vast majority of customers will return to a normal service, barring any residual reconciliation required."
At Westminster DUP Strangford MP Jim Shannon asked for an assurance that the Government would intervene to resolve the problem at RBS, which is majority owned by the state.
The PM said RBS had told the Northern Ireland Secretary of State they would reimburse any customer for "anything that's incurred because of this" and added that "lessons must be learnt"
What happened isn't acceptable. It's clearly an operational matter for the bank but the financial services authority has been monitoring this very closely.
DUP parliamentary group leader Nigel Dodds said it was unacceptable that 15 days after the problem appeared, individuals, households and businesses still could not get access to their money in the normal way.
He asked what measures were being taken to provide "a bit more flexibility for those who may be facing cash flow problems" and argued compensation must be paid in full in all circumstances.
Northern Ireland Office Minister Hugo Swire told Mr Dodds he will "certainly have the support of the Secretary of State and myself in ensuring that no one loses out as a result of this IT failure. I was specific on that point to the chairman of RBS."
He added he would check on Monday that the bank was "making the progress to clear up this sorry mess that they say they are going to over the weekend".
Meanwhile, Labour called on the Government to hold an emergency summit to help customers affected by the debacle.
Later DUP MP Nigel Dodds attended a cross-party delegation meeting at the RBS headquarters in London to put pressure on banking bosses.
Local MPS were told technical resources had been put in to deal with the problem, while extra resources may be sent to Northern Ireland ahead of the Twelfth holiday period.
"The Ulster Bank staff are under enormous pressure in the banks, they need the extra help in," Mr Dodds said.
"They said that they would certainly look at reviewing what they were going to do next week."
The MP said the banking group would take compensation on a case by case basis.
We have said to them, don't wait, start working on it now. Put the extra resources in now so people can start to plan for the future.
SDLP leader Dr Alasdair McDonnell described the meeting as "direct and constructive."
The South Belfast MP added: "People and businesses affected directly and indirectly by this crisis must be given a clear indication of what compensation they will receive, when these technical problems are going to be fixed and concrete assurances must be given that this will not happen again."
The IT problems that hit the bank have affected RBS and NatWest customers too and resulted in a huge backlog of millions of transactions.
But Mr Dodds called for a full independent investigation to determine whether the technical "sequencing" issue was to blame for NatWest and RBS accounts being prioritised or whether Ulster Bank customers "were left at the end of the queue".
In its latest statement, RBS said that the speed of bringing accounts up to date has increased "significantly" over the past few days.
It continued: "For example, over last weekend we cleared another 2 days of backlog and this week we have cleared payments which will allow customers to see a material improvement in the information they have.
"This compares to several days to clear a single days' transactions at the start of the incident.
"We will continue to provide updates daily on our progress."
Ulster Bank bosses will appear before the Department of Enterprise and Trade committee at Stormont on Thursday for questions.