Set of Finite Measure: Uncountable disjoint subsets of non-zero measure

No. Suppose $A$ is an uncountable disjoint union of measurable subsets $A_i, i \in I$ with positive measure. Then $I$ is a countable union of the sets of indices $i$ such that $\mu(A_i) > \frac{1}{n}, n \in \mathbb{N}$, so it follows that one of these sets must be uncountable. In particular a countable union of some subcollection of the $A_i$ has arbitrarily large measure; contradiction.