Deciding bound on Tensor Rank for a fixed value
Testing for constant border-rank can be done in polynomial-time, I'm pretty sure, if you're talking about tensors of fixed arity. More precisely, testing for border-rank $\leq r$ in $\mathbb{C}^{d_1} \otimes \dotsb \otimes \mathbb{C}^{d_k}$ can be done in something like $k(d_1 d_2 \dotsb d_k)^{2(r+1)}$ evaluations of $(r+1) \times (r+1)$ determinants plus some additional constant work (constant depending on $r$).

Theorem 3.7.1.1 of Landsberg's tensor book ("Inheritance") says that the equations for the variety of tensors of border-rank $\leq r$ in $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \dotsb \otimes \mathbb{C}^{d_k}$ are given by the equations for border-rank $r$ in $\mathbb{C}^r \otimes \mathbb{C}^r \otimes \dotsb \otimes \mathbb{C}^r$ together with all $(r+1) \times (r+1)$ minors of flattenings. Since $r = O(1)$, there are only finitely many equations of the former type.

Border-rank 1 = rank 1, and more generally rank $\leq r$ implies border-rank $\leq r$. But border-rank 2 can have rank anywhere from $2$ to the arity $k$, and border-rank $3$ in arity 3 can have ranks 3, 4, or 5. Beyond that I don't know about testing for rank.