More than 200 school and college leaders reported difficulties with the system and said staff and pupils could not book tests locally and were being directed to test centres long distances away, the Association of School and College Leaders (ASCL) said.
Several said home-testing kits were either unavailable or slow to arrive.
Parents have spoken of their frustration in not being able to access tests for their children.
Moz Bulbeck Reynolds, from West Berkshire, said she had been unable to book a test despite trying “almost constantly” since 9.30am on Monday through the government’s website.
She has been unable to send her nine-year-old daughter Matilda to school this week because of a lack of testing.
Having stayed at home last Thursday and Friday with cold symptoms, Matilda was refused entry to the school on Monday until she received a test, as per the local council’s rules.
“I feel sorry for my daughter… rejected at the school gate. It made me feel like a failure as a parent,” the 45-year-old said.
Schools minister Nick Gibb says schools will receive additional Covid-19 testing kits
The school has said Matilda either needs to be tested or quarantined for 10 days.
“I’m personally furious with the school for not telling me she needed a test when she did not have a high temperature,” added Ms Bulbeck Reynolds.
“I’m furious with the local council for moving the goalposts without informing the parents, and clearly I’m furious at the government for their incompetence.”
Another parent Samuel Austin, from Faversham, Kent, said he has been trying for 48 hours to book a Covid test after his young son started showing symptoms on Sunday.
“Everything we submit we get ‘this service is very busy, please try later’ or if we get through to finding a test site, we are told no site is available to see us,” the 35-year-old project manager said.
“The testing process is not fit for purpose. Regardless of the current uptake in test request, we should have been prepared.
“Ultimately my disappointment is for my son and my wife… I feel powerless to help them and it shouldn’t be this way.”
Video report by ITV News Political Correspondent Paul Brand
The situation has meant Mr Austin’s son cannot return to nursery and left his wife, a nurse at a local hospice, unable to work.
“As this hospice isn’t NHS she is not allowed to use the frontline channels for receiving a test,” Mr Austin added.
“Even though she has worked throughout the lockdown covering extra shifts where needed due to her hospice having the disease.”
Labour leader Sir Keir Starmer, who is currently self-isolating after a member of his family showed signs of coronavirus symptoms, said if schools can't get tests that is "a serious case of concern."
"The schools and colleges were closed down in March and therefore there has been six months to prepare for this and I think most families will be saying, they can't believe we're in this situation and the preparation isn't where it should be and I think the government needs to answer questions on this now," he added.
The problems in accessing tests meant that pupils and staff with symptoms were having to self-isolate at home for longer than necessary.
Schools and colleges, often with more than 1,000 students and staff, have each received a government supply of just 10 home-testing kits “to be offered in the exceptional circumstance where an individual may have barriers to accessing testing elsewhere”, the ASCL said.
One school leader said: “I have approx 10 pupils who are at home with symptoms all of whom are waiting for tests.
“Most have only been able to get postal tests, none have been able to get tests locally within seven days.
“Potentially I could have a number of positive cases linked to my school and not know it.”
Geoff Barton, General Secretary of the Association of School and College Leaders, said schools and college leaders “feel utterly” let down by the significant problems in accessing Covid tests.
“We are very concerned that the fantastic work of schools and colleges in putting in place a raft of safety measures in order to fully reopen for the autumn term is at risk of being derailed by a lack of capacity in the test and trace system,” Mr Barton said.
“Our frustration is with the government which has failed to live up to its promise to ensure that the test and trace system is able to meet the level of demand that it must have been perfectly obvious would be needed.”