  We address the problem of the additivity of the tensor rank. That is for two
independent tensors we study if the rank of their direct sum is equal to the
sum of their individual ranks. A positive answer to this problem was previously
known as Strassen's conjecture until recent counterexamples were proposed by
Shitov. The latter are not very explicit, and they are only known to exist
asymptotically for very large tensor spaces. In this article we prove that for
some small three-way tensors the additivity holds. For instance, if the rank of
one of the tensors is at most 6, then the additivity holds. Or, if one of the
tensors lives in $C^k \otimes C^3 \otimes C^3$ for any $k$, then the additivity
also holds. More generally, if one of the tensors is concise and its rank is at
most 2 more than the dimension of one of the linear spaces, then additivity
holds. In addition we also treat some cases of the additivity of border rank of
such tensors. In particular, we show that the additivity of the border rank
holds if the direct sum tensor is contained in $C^4 \otimes C^4 \otimes C^4$.
Some of our results are valid over an arbitrary base field.
