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3 year oldCovid-19 units in all but two of Brazil's 27 states are at or above 80% capacity, according to Fiocruz.
In Rio Grande do Sul state there are no intensive care beds available at all.
The warning came as the country registered its highest daily death toll yet with 2,841 dying within 24 hours.
That figure constitutes a large jump from the previous high of 2,286 on 10 March.
"The analysis by our researchers suggests it's the biggest collapse of the hospital and health service in Brazil's history."
Health officials in Brazil's most populous state, São Paulo, which on Tuesday also registered a record daily death toll, have called on the new health minister to consider imposing a national lockdown.
Marcelo Queiroga - who will be formally appointed as health minister later on Wednesday - is the fourth person to hold the office since the pandemic began.
He was given the job on Monday by President Jair Bolsonaro, who has faced widespread criticism over his handling of the pandemic.
President Bolsonaro has consistently opposed quarantine measures introduced by state governors, arguing that the collateral damage to the economy would be worse than the effects of the virus itself.
In remarks to the media on Tuesday, Mr Queiroga urged Brazilians to wear masks and wash their hands but stopped short of endorsing a lockdown or even social distancing measures.
The cardiologist told CNN Brasil that while "lockdowns were used in extreme situations, they could not be government policy".
That drew a strong response from João Gabbardo, the head of Sao Paulo's Covid-19 emergency body.
Posting on Twitter, Mr Gabbardo said private hospitals had been requesting space in the public health system because of the demand for intensive care beds. "When he [Queiroga] takes over, he will face the worst numbers in the pandemic," Mr Gabbardo tweeted, adding: "Suggestion: do not take a stand against a national lockdown."
President Bolsonaro has consistently played down the dangers of the pandemic - last week telling people to "stop whining" about Covid-19.
Brazil has the second highest number of infections and deaths in the world, behind the US. In total, the country has registered more than 11.6 million infections and 282,000 deaths since the pandemic began.
The latest surge in cases has been attributed to the spread of highly contagious variants of the virus.
The government has also faced criticism for the slow rollout of vaccines. It is currently distributing the Oxford-AstraZeneca and Chinese-developed CoronaVac jabs and has placed orders for the Pfizer-BioNTech, Johnson & Johnson and Russian Sputnik V vaccines. So far about 4.6% of the population have received at least one dose.
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