Peter Miller told the union's annual conference in Torquay that some schools were so short of money and qualified staff they could be forced to send children home for an afternoon a week. And schools across the country are unable to teach the full national curriculum, he said.
Mr Miller called for an additional Pounds 3 billion of education investment, to bring Britain's spending in line with the levels of the mid-1970s. He also demanded much more help for schools whose pupils were socially deprived.
"I've had half a dozen heads and deputies come to me and say they have refused pay rises from governing bodies because of the financial difficulties their schools are under," said Mr Miller, who is deputy head of the Wrenn School in Wellingborough, Northamptonshire.
"We have heard stories from various schools saying, 'We can't provide full-time education for all of our pupils.' They're facing a prospect of sending a year group home for one afternoon a week.
"This could be the year we finally reach meltdown. We're staring it in the face."
Schools have been pressed even harder by the crisis in teacher recruitment in recent months.
"We have got the problem of the gold-rush people going for early retirement, which means over the next two or three years this is going to be the big issue," said Mr Miller.
"People aren't being attracted into teaching: there's a general lack of esteem for the profession."
Schools, too, are finding themselves unable to offer shortage subjects such as technology, languages and science.
"There are certainly schools that can't afford to do the technology that they should do," said Mr Miller. "You need small teaching groups, but they haven't got the resources.
"We can't see how a number of schools are applying the full curriculum. This is true for science teaching and languages and, quite possibly in the near future, with maths and the core subjects. Schools are already putting people in front of the class who are not really qualified. How are these people going to motivate demotivated children?" Mr Miller attacked the "intrusive, disruptive" inspection system and argued instead for school self-evaluation.
He was also critical of a "shameful backlog of maintenance work" that is leaving children in uncivilised conditions and sceptical of Labour's plans to repair crumbling schools with private money.
Mr Miller called for a further slimming-down of the national curriculum to give teachers greater freedom.
He also floated plans for teacher training to be reformed. Educational psychology and philosophy should be reinstated as important components of training courses, he said. Trainees should spend a further two years of apprenticeship in school on a reduced workload, and maintain their links with higher education.
In his own conference address, the association's general secretary, John Sutton, warned that without new education spending, the national curriculum would be impossible to deliver while Sir Ron Dearing's recommendations for 14-19 qualifications would "drain away into the sands" because the nation was not prepared to pay.
He described chief inspector Chris Woodhead as "absurd" for suggesting that funds can be transferred away from secondary schools to the primary sector.
"We would love to know what surplus capacity he has identified," said Mr Sutton. "But that would be asking him to back up his statements with hard evidence. It's probably too late to ask him to adopt new habits."
Secondary Heads Association wishlist for a new administration
* adequate funding for secondary schools and FE colleges
* a general teaching council with real powers
* a unified 14-19 curriculum structure
* extensive improvements to school buildings
* special support for schools in disadvantaged areas
* an inspection system integrated with school self-evaluation and review
* a clear role for LEAs in admissions and strategic planning
* no extension of selection
* more early years and nursery education
* a solution to the forthcoming crisis in teacher supply