Between November 19 and November 23 more than 1,300 people received positive results, despite the tests being void.
Duncan Larcombe, a former royal editor at The Sun newspaper, said his 14-year-old daughter was among those who received an incorrect result.
The PR boss from Maidstone, Kent told BBC News his family took it “very seriously” and both his children were sent home from school to self-isolate.
Mr Larcombe said the incident, which prevented him from working, brought into question whether the testing system was adequate.
He told the broadcaster: "The entire economy is relying on the competence of the testing laboratories and if they are not doing their job they need to be held to account."
His daughter was able to get another test on Thursday which was negative, but given Kent has just been put into tier 3, Mr Larcombe questioned if the government’s "modelling is flawed".
A Department of Health and Social Care (DHSC) spokesman said this was an “isolated incident” and that all those impacted will be asked to take a further test.
They added it was caused by an "issue with a batch of testing chemicals" and that the problem would be fully investigated.
The DHSC did not comment on whether the error affected regional figures of infection rates.
Test and Trace has been beset with issues, with figures earlier this week revealing about four in 10 contacts of those who test positive for the virus are still not being reached.
The Prime Minister announced an extra £7 billion for Test and Trace in his winter plan on Monday, in a bid to increase testing and improve contact tracing.
It takes the overall funding provided for Test and Trace this financial year to £22 billion — nearly a fifth of the entire annual NHS budget — which has been met with some criticism.
But Boris Johnson was adamant the system provided "value" for money although admitted there had been "teething problems".