  Schlichting conjectured that the negative K-groups of small abelian
categories vanish and proved this for noetherian abelian categories and for all
abelian categories in degree $-1$. The main results of this paper are that
$K_{-1}(E)$ vanishes when $E$ is a small stable $\infty$-category with a
bounded t-structure and that $K_{-n}(E)$ vanishes for all $n\geq 1$ when
additionally the heart of $E$ is noetherian. It follows that Barwick's theorem
of the heart holds for nonconnective K-theory spectra when the heart is
noetherian. We give several applications, to non-existence results for bounded
t-structures and stability conditions, to possible K-theoretic obstructions to
the existence of the motivic t-structure, and to vanishing results for the
negative K-groups of a large class of dg algebras and ring spectra.
