Mateu and Orobitg proved (in Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990)) that given $\lambda > 1$ and $d - 1 < \alpha \leqslant d$ there exist constants $C$ and $N$ (depending on $\lambda$ and $\alpha$) with the following property: For any compact set $K$ in ${\mathbb {R}^d}$ one can find a (finite) family of balls $\{ B({x_i},{r_i})\}$ such that (i) $K \subset \bigcup {B({x_i},{r_i})}$, (ii) $\sum {r_i^\alpha \leqslant C{M^\alpha }(K)}$, ${M^\alpha }$ denoting the $\alpha$-dimensional Hausdorff content, and (iii) the dilated balls $\{ B({x_i},\lambda {r_i})\}$ are an almost disjoint family with constant $N$. In this paper we prove that such a result is false for $\alpha \leqslant d - 1$.