I'm going to go out on a limb here and guess your cards are running too hot... What sort of case are you using ? Fan speeds ? VRAM speeds ? Scrypt or SHA256 hashing ? Need more info.
The rig shutting down is probably due to hitting a thermal limit.
The card running too hot shouldn't be melting the power connections!
My best guess would be that not all the plugs were properly inserted and making contact with the pins. This would have caused more current to flow down the ones that were making contact, heating them up and then melting the connectors!
You need new connectors for that and make sure all the connections are properly connected.
Wow I think the word connection has now lost it's meaning; I've said it too many times!