  We study sets of local dimensions for self-similar measures in $\mathbb{R}$
satisfying the finite neighbour condition, which is formally stronger than the
weak separation condition but satisfied in all known examples. Under a mild
technical assumption, we establish that the set of attainable local dimensions
is a finite union of (possibly singleton) compact intervals. The number of
intervals is bounded above by the number of non-trivial maximal strongly
connected components of a finite directed graph construction depending only on
the governing iterated function system. We also explain how our results allow
computations of the sets of local dimensions in many explicit cases. This
contextualizes and generalizes a vast amount of prior work on sets of local
dimensions for self-similar measures satisfying the weak separation condition.
