There are cardinality $c$ Borel sets of finite ${\cal H}^s$ measure in ${\mathbb R}^n$. Assuming the Continuum Hypothesis, well-order these by the first uncountable ordinal as $E_\alpha$ for $\alpha \in A$, and define $X_\alpha = E_\alpha \backslash \bigcup_{\beta < \alpha} E_\beta$. Then $X_\alpha$ are disjoint Borel sets of finite ${\cal H}^s$ measure. For any Borel set $E$ of finite ${\cal H}^s$ measure, $E = E_\alpha = \bigcup_{\beta \le \alpha} (E \cap X_\beta)$ for some $\alpha$,
and ${\cal H}^s(E) = \sum_{\beta \le \alpha} {\cal H}^s(E \cap X_\beta)$ with $E \cap X_\beta = \emptyset$ for all $\beta > \alpha$.

Hmm, I'm not sure about the case ${\cal H}^s(E) = \infty$: I think it's true that $E$ has Borel subsets of arbitrarily large finite ${\cal H}^s$ measure, but I don't have a reference at hand and can't think of a proof. If this is the case, then the result is true.