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.