Scottish Labour said contact rates and times have fallen 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.
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 warning came as MP Ian Murray said he had been contacted by many 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.
Chief medical officer Dr Gregor Smith separately revealed Test and Protect workers were regularly receiving abuse from people who they instructed to isolate after coming in contact with a positive case.
The Test and Protect 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, Labour 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.
The figures were released as Scotland recorded one coronavirus death and 3,118 cases in the 24 hours to Tuesday.
The daily test positivity rate is 11.6 per cent, down from 12.6 per cent the previous day.
A total of 215 people were in hospital on Monday with recently confirmed Covid-19, up 13, with 20 patients in intensive care.
Speaking at Tuesday’s coronavirus briefing, Ms Sturgeon said: “Test and Protect obviously comes under much more pressure the higher case numbers are because there is more work to do.
"And we have a duty and an obligation to monitor that to make sure it has the resources it needs.
"So Test and Protect will have been working under much more pressure over the past week than it has done, at any point since probably the early part of this year. I'm not complacent about that.”
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.”
Anecdotal evidence has shown many people who expect to be told to isolate due to a close contact testing positive are not being contacted quickly enough.
In an indication of this, Mr Murray posted 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.”
Ms Sturgeon pointed to the most recent figures for the week up to June 20, where Scotland had outperformed the World Health Organisation (WHO) criteria of 80 per cent of new cases having close contacts traced and in quarantine within 72 hours of case confirmation, with 95.4 per cent.
She said: "So it's meeting that target, but we continue to assess. Obviously Test and Protect will also work as any organisation and system like that will do, they will flex their resources to make sure that they are meeting the greatest need."
"They're making much more use of text messaging, for example, so that somebody who's been asked to self isolate is getting that notification as quickly as possible.”
Also speaking at Tuesday’s briefing, Dr Smith said more Test and Protect staff were receiving abuse from people who they contact.
He said: “One of the features that we're occasionally seeing more often that has been reported from these teams is that when they're contacting the public and asking them to isolate, there are more often cases of people beginning to be abusive to them.
"I would ask people, please be patient with these teams when they contact you and listen to what they have to say. They are trying their very best to make sure that we're all remaining protected from Covid-19.”
A Scottish Government spokesperson said: “Our Test and Protect system is resilient and has continued to operate well over the last few weeks, meeting World Health Organisation criteria of closing 80 per cent of cases with 72 hours.
“We have taken steps to increase capacity within our contact tracing system, to account for the growing number of Covid cases. This has included making contact tracing scripts easier and increasing the use of our online contact tracing form. People are now sharing more information about their contacts and places they have visited digitally instead of over the phone.
“International travellers are treated differently according to the country they have travelled from. Those coming from red list countries have to enter managed isolation.
"Travellers from amber list countries are required to isolate at home and take three tests, minimising the risk of them transmitting the virus if positive. These travellers are still contacted by e-mail to remind them of the requirements.
"Green list countries travellers are required to still take a pre-departure test and a test on day two after arrival. This approach balances the risk of importing new cases with the need to reprioritise Test & Protect resources.”