Eisenbud and Harris introduced the theory of limit linear series, and constructed a
space parametrizing their limit linear series. Recently, Osserman introduced a new space
which compactifies the Eisenbud-Harris construction. In the Eisenbud-Harris space, the set
of refined limit linear series is always dense on a general reducible curve. Osserman asks
when the same is true for his space. In this paper, we answer his question by
characterizing the situations when the crude limit linear series contain an open subset of
his space.