Overcoming the reaction force From what I remember of mechanics, an object placed on the earth will exert a force on the ground called weight, and the ground will exert and equal and opposite force on the object called a reaction force. If the object is heavy enough, it will go through the ground! Is the reaction force becoming overcome? How can this be?
 A: 
Is the reaction force becoming overcome?

No, not in the sense what we mean by reaction. Action reaction pairs are always equal and opposite (and act on different objects). But it might suffice to say that the ground isn't able to produce sufficient reaction force to balance the weight of the object. 

How can this be?

I am interpreting this as:
"What is the proper explanation of this sinking phenomena?"
When we place an object on the surface of the earth then the first thing that happens is that the object creates a depression in it this is because atoms consist of charge particles and when they come quite close to each other (even though neutral at large distances) they apply force on one another which follows Newton's third law. This means that not only a depression is created in ground but also in the object itself, but for our purpose we assume that the object which we are placing on the surface of earth has infinite Young's modulus. Now as soon as this depression is formed a restoring force is generated which starts pushing the object upward and this force is proportional to the strain caused by the object. But it should be noted that this reaction only balances the gravitational force after certain amount of depression is produced and then and there the object comes to at rest. 
Now let's talk a bit about this restoring force.
Whenever a force is applied on an object a strain is caused. This strain is given by $\Delta \ell / \ell$ for longitudinal strain. Now the restoring force is proportional to the amount of strain produced (within the elastic limit of the material). This restoring force is generated because all materials are made up of atoms and when they are pushed together then they resist this by applying a force which originates from the EM force between the atoms. This restoring force is measured via stress ($F/A$, $A$ for area of cross section).

When the elastic limit is reached (here indicated by the yield strength) the proportionality relation ends and the material starts showing plastic behavior and in this region even a small increment in stress causes a large amount of strain. 
Now in case of ground when elastic limit is reached the ground starts deforming much faster and applies a much smaller reaction force as compared to the case if it was in its elastic limit. Now in the graph you can see that for each material there  is a point of ultimate strength after which a strain is produced even for smaller stress and then the material reaches it's fracture point where it applies no reaction force at all. 
So you can see that when the weight on the ground causes stress larger than ultimate strength then the ground starts collapsing and nothing else can be done to stop the object from sinking any further. 

This answer provides a beautiful microscopic explanation of the case. 
A: The other answers appear to be correct, but I just want to give a simple example to illustrate the point.
Consider a plank of wood, thin enough that you can punch through it.  If you just start to push on the wood, it should stay intact, and push back with an equal and opposite reaction to how hard you push it.
At some point though, if you push hard enough, instead of the plank pushing back with more and more force, it breaks.  You pushed against it with more force than it can push back.
This means that your hand moves right through the plank, being slowed down; but not completely stopped; because there is still an acceleration which was not totally resisted by the pushback of the wood.
The same thing can happen in theory with the ground, if the ground was too weak to cancel out the forces being applied to it.
Other answers elaborate on how this works from a material perspective, so I don't feel the need to go into that.
A: Think of the ground as a trampoline or silly putty. Since nothing is perfectly rigid, the deformation experienced will be a function of weight. This deformation manifests itself as molecules being pulled apart.
At some point when the weight is too much, or the ground is weak that pulling apart will be enough to completely sever the bonds of some molecules allowing for large separation or a crack to form. 
Alternatively the molecules slip past each other moving the material on either side of the object allowing for the object to sink.
