Where does this misconception come from? It comes from observing corpses and noticing nails that are consistently longer than they were before the corpse became a corpse. I'll spare you the gruesome images, but do a Google image search for fingernails after death and you'll find many images of corpses with long finger nails.
The truth? Fingernails stop growing at death. Why then do they look longer? Fingernails on corpses look longer because skin begins to shrink after one dies. This pulls back the skin at the edge of the fingernails, revealing more nail and giving the appearance of growing fingernails after death.
Now you know something you didn't know before reading this blog. :-)