Could something like this happen to Bitcoin?
In other words, are people starting to want to save their BTC coins so much that their value is lost?
I do not understand your question to be honest. Why would the value of Bitcoins be lost due to people wanting to save their Bitcoin? How can this apply to Bitcoin which is a digital Asset and not something you can melt?
People saving their Bitcoins only leads to scarcer Bitcoin, I do not see why it would become a bad thing unless nobody wants to spend their Bitcoin any more.
The melting of silver and metal when the intrinsic value is greater than the nominal
Was just an example to showcase how individuals hoard perceived better money
And push them out of the market and the bad remains in circulation.
I doubt if OP called it a bad thing, if people start holding it means they believe it's better to the alternative they would be spending.
But well the no value could mean more since its joker or it's a mistake.