NHS Lothian has urgently written to 242 women involved to apologise and offer appointments in the next three weeks.
The health board is also investigating a further 127 women who have moved to other areas, to check if they are also due for screening.
The women are considered at increased risk of breast cancer due to their genes or family history. Some of them should be screened once a year, while others are offered appointments every 18 months or two years, depending on the level of risk.
NHS Lothian discovered the women had not been invited for screening as scheduled after a review of patient records.
The incident is now being investigated. Women not on the increased risk list are not affected, NHS Lothian said.
Katie Dee, Deputy Director of Public Health at NHS Lothian, said, “Following an extensive review of participant records within our Increased Risk Breast Screening Programme, it was identified that a number of women may not have been invited for screening as regularly as they should.
“We are writing to the women who are affected to offer our sincere apologies, provide more information and to invite them for a new screening appointment.
“This incident is being investigated thoroughly and appropriate steps are being taken to ensure it does not happen again.”
Ms Dee added, “We apologise for this error and any worry it may cause. We have set up a dedicated helpline that can provide more information, advice and support. This is available on 0800 085 8531. If anyone has concerns, we encourage you to please get in touch.”
Public Health Minister Maree Todd said the Scottish Government is working with NHS Lothian to ensure the error is not repeated in another health board.
"The board is, rightly, apologising to all of those affected and we are clear that steps must now be taken to ensure this is not repeated in NHS Lothian or any other board,” she said.
She added: “It is important to note this incident is not connected to the national Breast Screening Programme, which covers all women between the ages of 50 and 70.
“We are working with NHS Lothian to ensure all those affected have been identified and they will be notified as a matter of urgency. The board will also ensure that anyone who needs to be seen will be offered a screening appointment urgently.”
Dr Jodie Moffat, head of the early diagnosis programme at Cancer Research UK, said the incident was “worrying” as the women involved were at increased risk.
"Because the women affected by this error are at a higher risk of developing breast cancer, it is very worrying that some have not been called forward for their screening as regularly as they should. It's crucial that lessons are learnt from this and that it doesn't happen again,” she told The Times.