Consider a noncompact homogeneous space equipped with the action of a one-parameter diagonalizable subgroup. In this dynamical system we study the set of points with precompact orbit which, due to the work of Kleinbock–Margulis, is known to be a dense set of full dimension. Motivated by a theorem of Davenport–Schmidt in Diophantine approximation we revisit this work and give a construction for bounded orbits with large closures.