Good question. Actually, our wave ansatz remains correct for large amplitude oscillations as well, so long as the springs remain harmonic (i.e., as long as Hooke's law holds). However, if you remember how we derived Hooke's law from the interaction between atoms, what we actually did was to expand V(x) near its minimum to get a harmonic (parabolic) potential. Once the amplitude of oscillation becomes large, then we need to keep the higher order terms in the expansion. In this case, our solution breaks down.
[ This is an aside,.... if you think of the corrections to parabolic as being small (like you are asked to assume in the homework problem on thermal expansion in problem set 2) then this weak perturbation can actually be thought of as simply causing phonons to scatter from each other. In other words the wave ansatz (which quantized becomes the phonon) is almost the correct dynamics, but the small correction correct the solution which is equivalent to the phonon being scattered occasionally. ]
It is also the correct intuition that once the amplitude of oscillation gets sufficiently large, bonds will typically break and a solid will become a liquid. The history of this idea goes back to Fredrick Lindemann, the first Viscount Cherwell (the guy the lecture theatre is named for). He guessed that a solid should melt when the amplitude of oscillation reaches a certain fraction (about 10%) of the interatomic spacing. This is now known as the Lindemann criterion -- which works fairly well as a rule-of-thumb for predicting melting temperatures.