There are quite a few examples like this from quantum information theory (beyond the basic fact that 2-tensors are "easy" and 3-tensors are "hard", which I already mentioned in another answer). Here are a couple:

• A big problem in the field is, given a density matrix (mixed state) acting on $\mathbb{C}^n \otimes \mathbb{C}^n$, how can we determine whether or not it is separable? It turns out this problem is easy if $n = 2$: just compute the partial transpose and then compute eigenvalues (in particular, it's polynomial-time and just makes use of "standard" linear algebra tricks). By contrast, when $n \geq 3$, it is known that the set of separable states is not semidefinite representable (loosely: it's not possible to use "standard" linear algebra tricks to determine separability).
• How well can you distinguish two quantum states? If there are $n = 2$ states, there's an explicit formula for the maximum probability of distinguishing the states, and the measurements you should perform to distinguish them as well as possible come from eigenstuff (see Nielsen and Chuang's book, for example, or basically any book on quantum information theory). By contrast, when there are $n \geq 3$ states, there is a semidefinite program that can compute the optimal distinguishing probability and the optimal measurements, but there is no known explicit/closed-form formula for it (and as far as I know, it's expected that no closed-form-ish formula exists).