Max load supported by a single pin of ATX 24-pin connector
The datasheet (it's a great thing to find such details) of the connector tells us:


  Current = 13A max. per circuit when header is mated
  to a receptacle loaded with 45750 Mini-Fit® Plus HCS
  Crimp Terminal Crimped to 16 AWG wire., See Molex
  product specification PS-45750-001 for additional
  current de-rating information.


The advice of Molex in their additional product specification actually shows, that on a fully populated 24 pin Wire-To-Board connector, the rated current is decreased to 8A per pin:



I wouldn't go anywhere near the maximum on a single pin though, at that current they will get hot (specified to a temperature rise of 30K above ambient at rated current).

So if you spread out the load over several of the 24 pins, you should be fine.

A low quality connector might not hold up to those specs - another reason why not to go to the limits.