Why doesn't the water spill out? In this experiment, a number of coins are put into a cup full of water, without spilling it.
http://www.youtube.com/watch?v=N2mKpZHnEzw
Firstly, let me clarify one thing.  

If you fill up a cup of water to the brim, in such a way that even another drop of water will cause it to overflow, can this cup take a coin, instead of a water-drop?
   - If no, then it doesn't matter and there's no point to this question.
   - If yes, then my question is this: why another coin and not another water-drop?

---This part exists only if the answer to the question above is yes--- 
They say it's because of surface tension, but I still don't get how that explains it. If the surface tension can hold together another coin, then why not another drop (which actually has less volume than a coin)?
The only reason I can think of is that the adhesion between the water and the coin is high, which sorta pulls the water molecules towards the coin, increasing the density of water immediately around the coin - thereby making up for the extra space the coin takes.
(But thinking about that, that doesn't seem to make much sense either. The coin is at the bottom, and I don't know if adhesive forces can produce such increases in density.)
So: what is the reason for this? Why a coin and not another water-drop?
 A: The real issue is that the cup wasn't really full so that adding anything more would make it spill.  You can clearly see the the level slowly growing above the top of the cup, as would be expected due to surface tension.  Eventually another coin finally exceeded the limit, and a little water spilled.  There is really nothing extraordinary going on here.
They could have just as well added some more water as individual drops and gotten the same effect.  You say that the cup was filled so that another drop of water would cause it to spill, but that was never stated nor demonstrated in the video.
Once difference between adding a water drop and a coin is that, if done carefully, the coin will cause less of a wave.  In fact the coin that caused the spill seemed to be added deliberately to cause a wave, like the person was tired of adding coins and wanted to see the spill already.  A coin can be inserted into the water edge-on, and cause a small wave when doing so.  A water drop will cause more of a wave because the surface tension of the water in the glass and that of the drop merge when they touch, which causes a sort of snap action that cause a wave.  This wave will more likely stress the miniscus at the edge to the breaking point than the tiny rise in water level due to the drop alone.
A: Edit: regarding your core question: no, if you drop a coin into a glass filled 'more' than the brim-level, such that any more water will cause it to spill, water will spill out.
But, instead of a coin, if you insert something that has other properties that allow it to absorb water whilst not suffering any volumetric changes and has strong cohesive force between it and water, it might be possible.

You may ignore the consideration of adhesive forces between water and coin for the purpose of explaining this effect.
It is surface tension: the strong cohesive forces that allow the water to reach heights above the brim of the glass is responsible for the effect.
In surface tension perspective, the external force in this experiment is gravity which attempts to 'flatten' out that 'bump' of water over the brim by pulling all the molecules downward.
Side experiment to confirm this: add soap to water to affect surface tension and reduce cohesive forces in the medium (water+soap solution). You can't add as many coins.
Related material:
[1] http://physicscentral.com/experiment/physicsathome/h2o-surface-tension.cfm
[2] http://ysp.wustl.edu/KitCurriculum/SurfaceTension/Surface%20Tension-Teacher.pdf
