$2\times2$ matrices are not big enough

Any two rotation matrices commute.

I like this one: two matrices are similar (conjugate) if and only if they have the same minimal and characteristic polynomials and the same dimensions of eigenspaces corresponding to the same eigenvalue. This statement is true for all $n\times n$ matrices with $n\leq6$, but is false for $n\geq7$.

The irreducible representations of $GL_n(\mathbb F_q)$ are well understood for $n=1, 2$, but there is still a lot that isn't known for $n \geq 3$.