The party said contact rates and times have declined over recent weeks as recorded cases increased to an all-time high. On Monday, 3,285 new cases were recorded – the highest ever daily figure.
Labour warned that Test and Protect had struggled to keep up with this increase.
The warning came as Scottish Labour MP Ian Murray said he had been contacted by hundreds of parents in his constituency whose children had been in class with someone who had tested positive, but had not been contacted by Test and Protect.
The proportion of Test and Protect cases closed within 24 hours have halved from a peak of 80 per cent in early April to 40 per cent this month.
The service aims to contact everyone who may be deemed a “close contact” of anyone who has received a positive test result. They are then asked to isolate for ten days since their last contact with the infected person.
The most recent Covid-19 Statistical Report shows that a third of cases from that week had not completed contact tracing by the time of publication.
Meanwhile, the party said the contact rate for those travelling from abroad has plummeted after the Scottish Government failed to provide the resources needed to cope with the new “traffic light” travel system, increasing the risk of importation of new strains.
In the first week of May, before the new system came into place, 92 per cent of those quarantining at home were contacted. By the second week of June this had fallen to just 43 per cent.
Mr Murray said on Twitter: “Huge amounts of correspondence from parents today who have children who have been in class with someone who tested positive for Covid, but have not been traced via the Test & Protect system. Hundreds of pupils and teachers have spent all weekend mingling and still no information.”
One Twitter user, Sian Aitken, said: “Yes a boy my son plays football with tested positive on a Saturday two weeks ago. Test & Protect didn’t contact the close contacts until late afternoon on the Tuesday so the children had been at school two days.”
Scottish Labour’s health spokesperson Jackie Baillie said: “Once again our Test and Protect system is reaching breaking point.
“Staff are working tirelessly to keep on top of the growing number of cases, but the system simply cannot cope.
“It did not come as a surprise to anyone that cases would increase as restrictions lift. The government’s failure to prepare for this is a downright dereliction of duty.”
She added: “If only the SNP spent as much time trying to improve these dismal figures as they do deciding how to spin them. Having a robust Test and Protect system in place is how we contain the virus in the future.
“We urgently need to put the resources in place to make sure Test and Protect is able to cope and bring the spread of the virus under control.”