A man who was told to drive to his nearest testing centre in Shropshire, with his kids aged one and three, was turned away after driving two hours to get there.
Steve Hyde arrived at the Telford testing site in time for his pre-booked slot, only to be told that the site was closed after it had run out of tests.
Mr Hyde told GMB that he drove miles across the country to get there after his two young children had coughs and therefore he "was trying to follow Government advice".
However he was left disappointed and said "the more that people struggle to get tests, the less that people are going to take advantage of it".
Mr Hyde told Piers Morgan and Susannah Reid that he had a personal reason for wanting to get himself and his children tested as soon as possible, as he had lost his father to coronavirus earlier in the year.
He said: "the saddest thing about it was not being able to be with him when he passed away".
He also said he fears that "the incompetence that I see from this track and trace system is going to lead to more deaths".
In response the Health Secretary, Matt Hancock, has said that "90% of people have to travel less than 22 miles and the average distance to get a test is under 10 miles."
He added: "There has been an increase in people coming forward to get tests who don't need a test, who haven't got symptoms and aren't part of one the programmes, for instance in social care."
He said: "If you have symptoms you should come forward and get a test, and the vast majority of people get one quickly and close to home".