Terminal Block Strip Electrical Block Connector 3A 5A 10 Amp /36 Wire Connectors min and max current limit

The voltage rating is related to the breakdown voltage of the plastic between the pins. You shouldn't exceed the breakdown voltage even of theres no current at all.

The current rating is, as you say, relates to the pin resistance and how much the connector will heat up. You shouldn't exceed the current rating even at very low voltages.

Edit:

As KellenJB says in another answer, it looks like the key thing you're missing is that the power consumed in the connector (and so the self-heating which could damage the connector) is not related to the voltage between the pins, but to the current through the pin. This current, combined with the (very small) resistance of the pin or contact, generates a small voltage between one end of the pin and the other (or between one pin and the socket its mated to). This voltage, multiplied by the current, gives the heat generated in the connector.

Because of the variation in contact resistance between mated pin/socket sets, two sets in parallel carrying twice the current meant for one will divide the current in inverse proportion to their resistances, with the higher resistance set shedding current the low resistance one must carry.

For example, with a 20 ampere source feeding a 10 milliohm resistance in parallel with a 20 milliohm resistance, 1/3 of the current (6.7 amperes) will flow through the higher resistance, while the remaining 13.3 amperes will flow through the lower resistance, exceeding the 10 ampere current rating of the low-resistance contact set.