Some school students were midway through the latest round of tests when they lost connectivity, while others were unable to login to the test at all.
Australian Education Union president Correna Haythorpe said on Wednesday the worst affected states were South Australia and Victoria.
"We believe it's time to have a comprehensive review of NAPLAN and that is also in terms of the way it is delivered," she told ABC radio.
Victoria's Education Minister James Merlino said the Federal Government was put on notice last year to get the system up to scratch.
"The rollout and performance of NAPLAN online is simply not good enough and Victorian students, parents and teachers deserve better," he added.
"The national bodies that run NAPLAN assured us the online test would work this year. But they have failed our schools yet again."
Connectivity problems were experienced across the country and are understood to be linked to the NAPLAN national firewall.
Outages also affected students in Western Australia according to the state's Education Director General Lisa Rodgers.
"I'm not going to discount these comments at all. These are very big issues and we're getting the calls from schools in terms of support in regards to access to the tests," she said.
Victorian schools and students participating in the online test can switch to paper tests if needed, Merlino said.
NAPLAN results for each school are published on the My School website, that helps parents and educators assess the performance of schools.