The stark warning was made this week by the Association of Colleges which says principals are being told by their local learning and skills councils that there is no available funding next year for recruiting extra students aged 19-plus.
The councils say that extra 19-plus recruitment is not a priority for funding unless it is for basic skills.
David Gibson, the AoC's chief executive, said: "When colleges are applying for extra funding for post-19, they are being told that no growth is being allowed."
The Department for Education and Skills wants the number of adults with level 2 qualifications - five GCSEs at grades A*-C or their equivalent - to be increased by a million by 2006.
Other key targets include raising the number of 19-year-olds with level 2 and 3 qualifications and improving the literacy and numeracy of 1.5 million adults with poor basic skills.
But many principals believe that if growth on courses not focused on the Government's priorities is to be achieved, colleges face stark choices.
They can either charge students and employers, withdraw offers of places, try to find funding from other sources, or go bankrupt.
Tim Jackson, principal of Sparsholt College, near Winchester, had been braced for problems with an application for funding to increase the number of full-time, 19-plus students by 45.
He had been told by the local LSC that "there was a difficulty nationally on the availability of post-19 funding as a result of having to prioritise". He said: "It is terrible if colleges are having to turn round and tell students, 'Sorry we can't accept you because we have run out of money'."
Sara Mogel, principal of West Cheshire College, said she expects to take on between 130 and 160 19-plus students next year, even though she has received funding for just six. "It's better than nothing, but only just," she said.
"We are not going to turn students away. How do you choose who can come and who can't?"
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