Schlichting conjectured that the negative K-groups of an abelian category vanish and proved this for noetherian abelian categories. Neeman's theorem of the heart, proved by Barwick, relates the algebraic K-theory of a derived category equipped with a t-structure to that of its heart, an abelian category. In this talk we will introduce algebraic K-theory, discuss recent joint work with Ben Antieau and Jeremiah Heller on the analogue of Schlichting's conjecture in the setting of derived categories, prove a version of the theorem of heart for nonconnective K-theory, and show more generally that the existence of bounded t-structures are obstructed by the nonvanishing of negative K-groups.