The phenomenon you're discussing sounds like what's called "multiple equilibria of the thermohaline circulation". This is currently a hot topic in the climate community, and a matter of much debate. As a result, I cannot give you a definite answer to your question. We can't yet say for sure if what you describe will happen, much less when.
The situation is this: currently, the water in the far North Atlantic is cooled by the atmosphere, and sinks because cold water is more dense than warm water. It flows southward deep beneath the surface, eventually rising to the surface in the tropics. Warm water travels north at the surface to replace the cold water which left. This warm northward-flowing surface water keeps the Atlantic and Europe relatively warm.
But computer models of atmosphere and ocean show that another situation is possible. If the polar waters are fresh enough, they cannot sink below the more salty tropical water, no matter how cold they get. Instead, the dense salty tropical water sinks, travels north, and rises in the polar regions. This flow is much weaker, so less warm water is carried north than in the previous case, and so the Atlantic and Europe are relatively cold. The atmosphere becomes stormier, and snow- and rainfall in the northern latitudes is increased: this maintains the fresh polar waters. Terminology note: since the flow in this reversed state is weaker, the phenomenon is often called a "shutdown" rather than a "reversal".
A large group of scientists, led by Wally Broeker at Columbia University, believe that switches between these patterns play a key role in the beginning and ending of the Ice Ages. Most scientists believe that the fairly regular cycle of ice ages is driven by small changes in the the orbit and spin of the Earth. These cycles lead to ice ages about every 100,000 years. However, there is increasing evidence that the extremely rapid beginning and end of ice ages is caused by reversals of the thermohaline circulation. Adkins and Boyle (1997) have found evidence of rapid changes in deep ocean currents by studying the sediment on the ocean floor.
When will we have another ice age? It's impossible to say at the moment. The present warm period has lasted about as long as the warm period (the Eemian period) which preceded the most recent ice age, and the orbital parameters described above are similar now to conditions at the end of the Eemian, so perhaps we're about due for a change soon. By "soon", I mean within 10,000 years, so don't get too worried. However, humanity's current impact on climate, through greenhouse gases and land use change, has never happened before, so it's hard to predict based on past occurrences.
In fact, paradoxically, global warming caused by humans may cause ice-age like climate! Broeker and others claim that as the Earth warms, we may make the surface waters of the North Atlantic unusually warm. This will make it more difficult for those waters to sink. If warming increases too much, it may cause a reversal of the thermohaline circulation, leading to a drastic cooling at high latitudes and a warming in the tropics. Several researchers, including Stocker and Schmittner (1997) and Manabe and Stouffer (1993), have taken a computer model of the atmosphere and ocean and added carbon dioxide to it at a rate similar to what we predict humans will do over the next century. After some delay, the thermohaline circulation in these models did reverse, resulting in drastic climate change.
At the moment, the idea that the thermohaline flow can reverse is pretty solid, I think. However, the ideas that such a reversal has been part of previous ice ages, or that it will happen if global warming continues, must be considered a strong hypothesis, with some data supporting it. There is still lots of debate and uncertainty, which we will resolve as time goes on.
References:
Hartmann, Dennis. Global Physical Climatology. Academic Press,
1994. Chapter 8 in particular.
"Earth's
past offers chilling global warming scenarios". Boston Globe, Dec 2,
1997
"Leading
Climate Change Expert Warns of Possible Collapse of Earth's Ocean
Systems", Lamont-Doherty Earth Observatory press release, Nov 27,
1997
Broeker, W. S. "What
If the Conveyor Were to Shut Down? Reflections on a Possible Outcome of the
Great Global Experiment" GSA Today, January 1999
Try the links in the MadSci Library for more information on Earth Sciences.