So if you look at the schematics, you can see the fan power pins are wired straight to the main +12V supply rail. This in turn is normally provided through either a power plane or a fat trace, meaning the power limit will be the fan header itself -- in fact, given the fan power header is comprised of standard stake pins, the power limit is more precisely in the fan
connector itself. Many of the cheapest Chinese fan connectors appear to use non-soldered (crimped) connections to what seems to be horse hair instead of copper wire, meaning I would be very concerned about pulling even an amp though them.
If you can get a high quality connector, the fan headers themselves are likely good for ~5A. Problem is, if your connector melts down / catches fire, it will damage the header and the PCB, and that would not be covered under warranty, so please be sure of the quality of the mating connector before trying to pull 5A...