Accepted for publication by the *Ann. Acad. Sci. Fenn.* Available on the arXiv. This is a joint work with John Osborne. For a transcendental entire function $f$, we study the set of points $BU(f)$ whose iterates under $ f $ neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if $U$ is a Fatou component that meets $BU(f)$, then most boundary points of $U$ (in the sense of harmonic measure) lie in $BU(f)$. We prove this using a new result concerning the set of limit points of the iterates of $f$ on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.