  We show that various methods for explicitly building resolutions of unbounded
complexes in fact fail when applied to a rather simple and explicit complex. We
show that one way to rescue these methods is to assume Roos (Ab.4$^*$)-$k$
axiom, which we adapt to encompass also resolutions in the framework of
relative homological algebra. In the end we discuss the existence of model
structures for relative homological algebra for unbounded complex under the
relative (Ab.4$^*$)-$k$ condition, and present a variety of examples where our
results apply.
