When reading a connector’s, datasheet, there’s almost always a current rating given for that connector. Can the connector withstand more current than this current rating?
The answer, in most circumstances, is no. This current rating is the connector contact’s current limitation, i.e. the maximum current the connector contacts can withstand. Please note that current through the contacts shouldn’t exceed this maximum current rating even at very low voltages - you cannot undervolt a connector to get more current out of it. Doing so will cause the connector to melt or the contacts to weld, both of which will destroy your connector.
The current rating of a contact is designed to meet the wire AWG current rating for crimping or soldering. So there’s usually a couple of choices on the wire size and current rating to apply for the same connector.