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The concept of the index of nilpotency is a pivotal one when dealing with nilpotent matrices. In basic terms, the index of nilpotency for a matrix refers to the smallest positive integer 'k' such that the matrix raised to the 'k'th power equals the zero matrix. Mathematically, for a matrix 'A', if we find that

\[A^k = 0\]
and for all integers 'j' less than 'k', \[A^j \eq 0\],
then 'k' is the index of nilpotency of matrix 'A'. This property is central to understanding the structure of nilpotent matrices since it provides a measure of how 'close' a matrix is to being the zero matrix in terms of its powers.

In the provided exercise, understanding that similar matrices of the same size can only be similar if they are raised to the same nilpotent index, 'k', helped in proving a given property for 3x3 matrices. But where does this principle falter? When we move to higher dimensions, as seen with the 4x4 matrices, we encounter matrices that might have the same index of nilpotency and yet not be similar. This highlights an essential point: while the index of nilpotency is an important factor, it is not the sole decider of similarity in higher dimensions.

When we talk about similar matrices, we're referring to a relationship where one matrix can be transformed into another through a specific change of basis. In other words, two matrices, 'A' and 'B', are similar if there exists an invertible matrix 'P', such that

\[P^{-1}AP = B\].
This operation can be likened to changing the 'viewpoint' from which we are examining the matrix, much like changing coordinates in geometry. Similar matrices share many properties, including eigenvalues, trace, and determinant, and hence can often be considered to be different expressions of the same linear transformation.

As elucidated in the exercise's solution, for the 3x3 nilpotent matrices, having the same index of nilpotency implies similarity. However, with 4x4 nilpotent matrices, the concept isn't as straightforward. This intricate topic requires a deeper understanding of the underlying structure of matrices, especially with respect to their Jordan canonical form.

Diving into an advanced topic like Jordan canonical form brings us to the heart of understanding the structure and classification of matrices. The Jordan canonical form of a matrix is essentially a block diagonal matrix that represents its most simplified version under similarity transformations. Each block, called a Jordan block, corresponds to an eigenvalue and is a nearly diagonal matrix with the eigenvalue entries on the diagonal and ones on the superdiagonal.

To put it clearly, every square matrix is similar to its Jordan canonical form, which is unique up to the order of its Jordan blocks. This form is particularly handy while working with nilpotent matrices since for nilpotent matrices, all eigenvalues in the Jordan form are zero. The arrangement of these blocks grants us crucial information about the similarity of matrices.

The last part of the exercise showcases that even with the same index of nilpotency, the number of nonzero blocks in the Jordan form can vary, fundamentally affecting the similar structure of nilpotent matrices in larger dimensions, such as 4x4 matrices. The Jordan canonical form thus acts as a more detailed lens for distinguishing matrices beyond what the index alone can reveal.