Let $A\subset \mathbb{N}$ be an additive basis of order $k$. Let $B=\{b_1<b_2<\cdots \}$ be the set of integers which are the sum of $k$ or fewer distinct $a\in A$. Is it true that $b_{n+1}-b_n=O(1)$? (Where the implied constant may depend on both $A$ and $k$.)