If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre Γ (Y) of Y as cofibre(hocolim_D (Y) → hocolim_C (Y)). We construct a comparison map \hatΓ_Y : holim_C Y → Hom (Z, \hatΓ (Y)) to a mapping spectrum of a fibrant replacement of Γ (Y) where Z is a simplicial set obtained from C and D, and characterise those poset pairs D ⊂ C for which \hatΓ_Y is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to Z. For example, if C is a finite polytopal complex with |C| ≅ B^m a ball with boundary sphere |D|, then |Z|≅_PL S^m, and \hatΓ(Y) and holim_C (Y) agree up to m-fold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for \pi_*(Γ(Y)) with E₂-term involving higher derived inverse limits of \pi_* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.