It is an encased dime that was irradiated at Oak Ridge National Laboratory in Tennessee, one of the primary sites for the Manhattan Project during WWII. After the war, the American Museum of Atomic Energy was opened to the public and the dimes were sold as souvenirs.
The radioactivity doesn't disappear altogether, but it gets exponentially smaller very quickly (it halves every 22 seconds, was it?). Even just a little while after it was irradiated, it was practically not radioactive.
True. If the entire 2.5 grams of the coin had been completely turned into Ag 110 by the neutron bombardment (and in reality only a tiny amount was) after just 1 hour only 2.13 X10^-49 grams would be left. That's a decimal point followed by 49 zeros then 213. That's just 163 half lives. This coin is possibly over 800K half lives old.