Summary: For a given cardinal \(\lambda\) and a torsion abelian group \(K\) of cardinality less than \( \lambda \), we present, under some mild conditions (for example, \( \lambda=\lambda^{\aleph_0}\)), boundedly endo-rigid abelian group \(G\) of cardinality \(\lambda\) with \(\operatorname{tor}(G)=K\). Essentially, we give a complete characterization of such pairs \((K, \lambda)\). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existence problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals \(\lambda> 2^{\aleph_0}\) for which there is a co-Hopfian abelian group of size \(\lambda \).