Thermocouple shorting problem -- readings go to 0... but only some of them...

I suppose so, if the OP's "piece of metal" is one of the alloys that make up a type K thermocouple: nickel-chromium (90% nickel and 10% chromium) or nickel-alumel (95% nickel, 2% manganese, 2% aluminium and 1% silicon).