Sum of low rank tensors How high can the sum of $k$ low rank $m\times m\times\dots \times m$ tensors of rank $t$ be? Is there a good upper bound?

Rank $t$ means $t$ (but no more) linearly independent rows/columns. If those $t$ that are linearly independent in the first matrix are also linearly independent to $t$ of those independent in the second one, you get rank $2t$, and so on.
Therefore, your answer is $r_{MAX} = \min \{ m, kt \}$.
Without knowing anything more about these matrices, their sum will have rank anywhere between $0$ and $r_{MAX}$.
