Sir Keir said the prime minister was ignoring the current problems with access to tests in the UK.
In recent days there have been numerous reports of people being unable to get their hands on a coronavirus test, or being told to travel long distances to get one.
The Labour leader raised the example of a mother in London who had been unable to get a test for her unwell child and after a whole day of back and forth, she was told the nearest available test was in Swansea.
He said: "If people are being told to go hundreds of miles something is wrong, it's got a lot worse in the last week or two."
On Tuesday, the Director of NHS Test and Trace apologised for the lack of testing capacity.
The PM said he sympathised with people who were struggling to get tests but repeatedly refused to admit there was a problem with NHS Test and Trace.
He said the median journey to take a test was under 10 miles and most people could get a test if they needed one.
Sir Keir said: "This is basic stuff, people who have got Covid symptoms are very anxious about themselves, their children, their families and what to do, it means they can't go to work, they can't send their children to school, it matters."
Mr Johnson said there had been 17.6 million coronavirus tests carried out in the UK, the highest in Europe.
He also said current capacity was at 320,000 tests a day, up from 2,000 in March, and repeated his pledge to hit 500,000 by October.
The prime minister appeared to blame the recent rise in coronavirus cases on people who had not been following the government's guidance.
Mr Johnson also faced several questions about his plans to amend the Brexit withdrawal agreement.
Yesterday the Secretary of State for Northern Ireland, Brandon Lewis, admitted their plans to amend the agreement would be in breach of international law.
Mr Johnson said he expected "everybody in this country to obey the law".
In a stormy set of questions from the SNP leader in the House of Commons, Ian Blackford attacked the prime minister's Brexit plans and accused him of attempting to take power away from Scotland.
Mr Blackford accused Mr Johnson of "creating a rogue state", and asked "why does the prime minister think he and his friends are above the law?"
After the prime minister said the new bill would protect the United Kingdom Mr Blackford accused him of lying.
Speaker Lindsey Hoyle intervened and the remark was eventually withdrawn.
Labour MP Siobhain McDonagh asked the prime minister to condemn the treatment of the Uighur Muslims by the Chinese government.
The prime minister said she was right to address the issue and he would raise it at every available opportunity.