The answer to the question is negative. Körner in Hausdorff dimension of sums of sets with themselves and Schmeling-Shmerkin in On the dimension of iterated sumsets showed that for any increasing sequence $\alpha_1 < \alpha_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim_{H} k A = \alpha_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.
See also the recent paper Dimension growth for iterated sumsets for some conditions under which we have dimension growth $\dim_H (A + A) > \dim_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim_H k A = 1$, much less providing quantitative bound on $k$.