Some amount of saving is good since the saved reserves can provide a buffer which can prevent short term perturbations from kicking off downward spirals.
If everyone saved every penny they don't spend on essentials, then that would stagnate growth if the savings weren't being put to work in meaningful ways.
Where we've been headed for the last 2 or 3 decades has been not only very low saving rates, but many people leveraging themselves (aka borrowing) further and further to live beyond thier means for a time. This is great for short term gowth, but can be dangerous since at some point too many people lose first the ability to borrow more, then the ability to service their existing debts (which is where many analysts think we are very close to or are already at) and can no longer continue to spend. If too many people reach this condition too close together in time, it can cause much of "the system" to come down like a house of cards, since the original lenders often borrow against their oustanding loans to get more money to make the next round of loans, so that everything's fine as long as there aren't too many defaults or delinquincies, but if that happens it can lead to a serious cascade which can potentially take down the entire institution.