That is what happened when the dollar was unmoored from gold. Then gold's cash value floated freely.
Quantitative easing has not resulted in increasing credit, and therefore the velocity of cash in the economy. If you make it painful to keep cash in one place they hope it is like a game of hot potato and the economy will move again.
The countering strategy is that there is nothing to invest in so it is better to create alternates to cash to preserve value
There are a number of problems with this. Some of them have been mentioned in passing in other comments, and most of them boil down to the disconnect between economic models of humans and real humans. Among other discrepancies, real humans have finite life spans and therefore a limited tolerance for variance; as such, under less than total economic security they generally refuse to dispense with saving if it's an option for them, even if the expected value of spending their savings is greater than not. So the real effect of progressively lower interest rates on monetary velocity diminishes as rates decline. A similar effect limits the ability of lower rates to increase the size of the money supply; people convinced that capital cannot be usefully employed in the present economic environment are unlikely to borrow regardless of how low rates are. A stagnant, highly unequal distribution of capital also limits the effects of lower rates, since the only people likely to be able to take advantage of them already have plenty of capital available to invest. Of course, the money supply side of this is somewhat less relevant, since no one is (yet) proposing negative lending rates, only negative deposit rates. However, negative deposit rates can actually shrink the supply of money, because the "confiscated" money ends up at the central bank where it would simply add to an already-gigantic pool of reserves that could be lent, but haven't been for the above reasons. That is, the money would essentially be removed from circulation and destroyed, potentially offsetting or even dominating any increase in velocity.
You can learn more about the relationships among these quantities by looking up "modern central banking theory" and basic macroeconomic theory in general.