We study the sizes ofδ-additivesets of unit vectors in ad-dimensional normed space: the sum of any two vectors has norm at mostδ. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. Füredi, J.C. Lagarias, F. Morgan, 1991). We show that the maximum size of aδ-additive set over all normed spaces of dimensiondgrows exponentially indfor fixedδ> 2/3, stays bounded forδ< 2/3, and grows linearly at the thresholdδ= 2/3. Furthermore, the maximum size of a 2/3-additive set ind-dimensional normed space has the sharp upper bound ofd, with the single exception of spaces isometric to three-dimensionall1space, where there exists a 2/3-additive set of four unit vectors.