Test and Protect, which calls people with a positive case of Covid-19 in order to trace who they have been in contact with, took a week or more to call 302 people who tested positive for the virus.
Public health experts have repeatedly highlighted the importance of a rapid and reliable contact tracing system in tackling coronavirus.
The system is intended to track down people who have been in contact of a positive Covid-19 case and are at risk of spreading the virus before they exhibit symptoms.
Figures, obtained through freedom of information requests to Public Health Scotland detail that 1,272 people waited four days between their case being logged in the Test and Protect system and their interview about contacts.
A further 566 people waited five days, 221 waited six days, and 302 waiting a week or more between August 2 and November 15.
Scottish Lib Dem leader Willie Rennie said the figures show Test and Protect must be expanded and called on the Scottish Government to “ramp up” testing.
He said: "People have made huge sacrifices this year to save lives and buy time. But these figures show that the Scottish Government must ramp up the testing and tracing capacity to meet the challenge of the new strain of the virus.
"Every minute counts when tracking down this virus. Ministers gave the impression that their system was superior, had all the staff it needed and could withstand whatever the virus threw at it. The First Minister repeatedly told me that Test and Protect was working well.
"In reality, we didn't have enough tracers to be able to go and hunt down the virus during the summer lull, or enough tracers to follow up leads when it took off again the autumn. The operation needed to be much bigger from the get go.
"Week after week I asked about preparing Test and Protect, the quarantine traveller spot check target that was missed for four months, and mass testing students to ensure their safe arrival at university. We know these events furthered the spread of the virus, contributing to the difficult position we are all in this Christmas.
"The Scottish Government's New Year's resolution must be to ensure that Test and Protect always has enough staff and expertise ready to deal with the massive challenge from this new strain of the virus.”
A Scottish Government spokesperson said: “We are continually reviewing our approach to contact tracing, and adapt and improve the system as issues are identified. The most recent preliminary data available, for the week 14-20 December, shows 97.9 per cent of all positive cases had their contact tracing completed within 72 hours, with 93.2 per cent completed within 48 hours.
“Our approach uses established, tried and tested contact tracing techniques. These are delivered by health protection professionals in teams within local NHS Boards with support provided by the National Contact Tracing Centre. The system will prioritise contacting individuals with a positive test as soon as possible, but it relies on people answering their phone. Some interviews may take longer to complete than others, for various reasons such as invalid telephone numbers and email addresses.
“If an individual has any questions after being asked to self-isolate after identification as a close contact of someone who has tested positive, they can contact the National Contact Tracing Centre on 0800 030 8012 for further advice.”