First-time grant applications are now being handled by Student Universal Support Ireland (Susi), a centralised service operated by the City of Dublin Vocational Education Committee (VEC).
But Union of Students in Ireland president John Logue revealed that 28pc -- or about 5,500 -- of the college-goers whose applications have been approved by Susi have yet to receive payment.
Dundalk Institute of Technology president Denis Cummins said he and other college chiefs have "real concerns about the capacity of Susi to cope".
He said there are already "issues around capacity" even though the system is so far only dealing with a small section of the student population.
"I think there is a lot of work to be done to regain confidence in the system," Mr Cummins said. Up to 150 students at Athlone Institute of Technology have been forced to accept free food boxes because they could not afford to feed themselves.
Trinity College Dublin and the Dublin Institute of Technology have started similar schemes.
Susi says "virtually all completed grant applications have been processed" by now.
Failure by students to provide necessary documentation has delayed most of the remaining payments, it says.
Some 1,500 more students are due to be paid this week, says City of Dublin VEC.
But Mr Cummins said: "I can confirm that there is real hardship by students."
He revealed that, at one third-level institution, there was an example of a student sleeping in their car in a car park, while another student had only €2 in their pocket to spend.
The City of Dublin VEC said that 3,000 students had still not provided their bank details or college registration confirmation, which was delaying their payments.
Some 27,606 applicants have already been awarded grants.