In a simple world, personal savings would have quite a lot to do with the level of household debt (from the Federal Reserve data discussed in this post), since debt might be paid down with savings, and people spending more than they earn are likely living in part on borrowings.
So I thought it might interesting to look at the ratio of the increase in household debt to personal savings. Note that, in the current revision of the series, personal savings has always been positive, but household debt has steadily increased until recently, when it started to decrease.
The ratio is above, and seems to date the explosion of borrowing and collapse in
Note that the real world is not so simple. Besides paying down debt, or borrowing to live, things can happen like borrowing to buy assets such as houses or stocks. This can increase your debt outstanding, even as you are saving. In essence, the debt is only one half of the household balance sheet. So I think the spike in 1999-2006 should be seen as household balance sheets blowing up in size (relative to savings) in both the asset and liability sides.
Unfortunately, when the party is over, the assets deflate, but the debt doesn't.