They have Aeromexico budget tickets, issued to airline staff and their relatives, which enable them to fly only if seats become available.
The airline says it advised ticket-holders not to fly during the peak holiday season.
Most are sleeping rough at Barajas, lacking funds to stay at Madrid hotels.
The cheap tickets they bought are not available to the general public.
Aeromexico rejected a plea from the 183 stranded Mexicans to send an extra plane to pick them up,
Spain's Efe news agency reported.
Dr Dulce Olveda, 65, told El Pais newspaper that there had been up to four Aeromexico flights but none of the stranded passengers had been given a seat. "People no longer have the funds to survive in the airport," she said.
Several passengers sent a group letter to the Mexican consul in Madrid requesting help, but embassy sources said they could not resolve "a recurring problem with this company [Aeromexico]".