A rise in the number children receiving Send support in schools has driven what the IFS calls a "dramatic and ongoing" increase in spending.
"The truly shocking thing is that despite this increase in spending, we are still left with a system that is failing to cope with the needs of these children and young people," Pepe Di'Iasio, general secretary of the Association of School and College Leaders, said.
One in five pupils (1.7 million) in England get some kind of support for special educational needs in school at the moment.
And 482,000 of them (or 5.3% of all pupils) get the higher level of support from individual EHCPs, a legal document setting out a child's tailored support needs, which councils have to provide and pay for.
The rising costs of Send support have forced local authorities into hundreds of millions of pounds of debt.
A government agreement meant councils have been able to keep Send deficits off their financial books, and now there are calls for the total debt - which is forecast to reach £5bn next year - to be wiped completely.
Amanda Hopgood, of the Local Government Association, says that would "put councils on a stable financial footing", as well as addressing "outdated legislation that leaves councils facing rising and unsustainable costs".
Munira Wilson, the Liberal Democrats' spokesperson on education, says councils have been "pushed to the brink", adding that "vulnerable young people are being set up to fail".
The IFS is predicting that the proportion of pupils with EHCPs will rise even further in the next four years, to 8% of all pupils aged four to 16.
In July, the government refused to rule out scrapping EHCPs as part of their upcoming reforms.
Education minister Georgia Gould has since said "there will always be a legal right to additional support", adding that children, families and teachers will be "at the forefront" of Send reforms.
Currently, the first £6,000 of Send support comes from school budgets - but costs above that in mainstream schools, and the full costs of support in special schools, are covered by councils.
Educating children in special schools is significantly more expensive than in mainstream settings. The average cost of funding one place at a state special school is £23,900 per year, or £61,500 per year for an independent special school.
The government wants more early intervention and more Send pupils to stay in mainstream education.
Angela Mawer, grandmother and legal guardian to seven-year-old Colby, says she felt like she had "won the lottery" by finding him a place in mainstream education which could cater to his needs.
"It's really sad that I should have to feel like that," she said.
"There isn't enough provision for children like Colby."