Actually, the paradox of thrift, which is one name for what you're talking about...
The paradox states that an increase in autonomous saving leads to a decrease in aggregate demand and thus a decrease in gross output which will in turn lower total saving. [...]
For although the amount of his own saving is unlikely to have any significant influence on his own income, the reactions of the amount of his consumption on the incomes of others makes it impossible for all individuals simultaneously to save any given sums. Every such attempt to save more by reducing consumption will so affect incomes that the attempt necessarily defeats itself. It is, of course, just as impossible for the community as a whole to save less than the amount of current investment, since the attempt to do so will necessarily raise incomes to a level at which the sums which individuals choose to save add up to a figure exactly equal to the amount of investment.
— John Maynard Keynes, The General Theory of Employment, Interest and Money, Chapter 7, p. 84
The theory is referred to as the "paradox of thrift" in Samuelson's influential Economics of 1948, which popularized the term.
... assumes a closed economy:
the paradox assumes a closed economy in which savings are not invested abroad (to fund exports of local production abroad). Thus, while the paradox may hold at the global level, it need not hold at the local or national level: if one nation increases savings, this can be offset by trading partners consuming a greater amount relative to their own production, i.e., if the saving nation increases exports, and its partners increase imports. This criticism is not very controversial, and is generally accepted by Keynesian economists as well, who refer to it as "exporting one's way out of a recession". They further note that this frequently occurs in concert with currency devaluation (hence increasing exports and decreasing imports), and cannot work as a solution to a global problem, because the global economy is a closed system – not every nation can increase net exports.
The two citations there are two (NYT) blog posts by Krugman.
Also, outside of strictly Keynesian perspectives,
particularly neoclassical economists, criticize this theory on three principal grounds.
The first criticism is that, following Say's law and the related circle of ideas, if demand slackens, prices will fall (barring government intervention), and the resulting lower price will stimulate demand (though at lower profit or cost – possibly even lower wages). This criticism in turn has been questioned by New Keynesian economists, who reject Say's law and instead point to evidence of sticky prices as a reason why prices do not fall in recession; this remains a debated point.
The second criticism is that savings represent loanable funds, particularly at banks, assuming the savings are held at banks, rather than currency itself being held ("stashed under one's mattress"). Thus an accumulation of savings yields an increase in potential lending, which will lower interest rates and stimulate borrowing. So a decline in consumer spending is offset by an increase in lending, and subsequent investment and spending. Two caveats are added to this criticism. Firstly, if savings are held as cash, rather than being loaned out (directly by savers, or indirectly, as via bank deposits), then loanable funds do not increase, and thus a recession may be caused – but this is due to holding cash, not to saving per se. Secondly, banks themselves may hold cash, rather than loaning it out, which results in the growth of excess reserves – funds on deposit but not loaned out. This is argued to occur in liquidity trap situations, when interest rates are at a zero lower bound (or near it) and savings still exceed investment demand. Within Keynesian economics, the desire to hold currency rather than loan it out is discussed under liquidity preference.
[The 3rd criticism, relating to the closed economy assumption, was already mentioned in the previous quote above.]
Although supposedly the "second criticism" is advanced by neoclassical economists, Wikipedia is citing an Austrian-school source (Roger Garrison) for that part.