Yes you can push 3A through some reputable brands of header pins and mating contacts, but that rating will assume the pin is soldered to a heavy ground plane, both parts of the connector are brand new, and probably that there is a 'R' in the month!
In a real world application, with the risk of elevated ambient temperatures, skinny traces, contacts that may have been mated/unmated repeatedly and have a little tarnish on them and possibly the female contacts may be off-brand (worst case: thinner metal and reduced contact force and area), as a rule of thumb, I wouldn't expect trouble with overheating under 1A.
However, it isn't usually the connectors burning up that's the big issue, its the voltage drop across them under load. If your ground pins are carrying significant current and if you have any TTL or TTL compatible CMOS on the board, or I2C stuff, their asymmetric logic thresholds make the logic '0' level particularly sensitive to ground offset and excess noise.
Worst case, if you are short of pins, you can use as many as possible for Ground and distribute power at a higher voltage on fewer pins, regulating it down to the logic Vcc using a local LDO regulator on each board so the positive supply voltage drop is compensated for.