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We prove a theorem on direct relation between the optimal fidelity ${f}_{\mathrm{max}}$ of teleportation and the maximal singlet fraction ${F}_{\mathrm{max}}$ attainable by means of trace-preserving local quantum and classical communication (LQCC) action. For a given bipartite state acting on ${C}^{d}\ensuremath{\bigotimes}{C}^{d}$ we have ${f}_{\mathrm{max}}{=(F}_{\mathrm{max}}d+1)/(d+1)$. We assume completely general teleportation scheme (trace preserving LQCC action over the pair and the third particle in unknown state). The proof involves the isomorphism between quantum channels and a class of bipartite states. We also exploit the technique of $U\ensuremath{\bigotimes}{U}^{*}$ twirling states (random application of unitary transformation of the above form) and the introduced analogous twirling of channels. We illustrate the power of the theorem by showing that any bound entangled state does not provide better fidelity of teleportation than for the purely classical channel. Subsequently, we apply our tools to the problem of the so-called conclusive teleportation, then reduced to the question of optimal conclusive increasing of singlet fraction. We provide an example of state for which Alice and Bob have no chance to obtain perfect singlet by LQCC action, but still singlet fraction arbitrarily close to unity can be obtained with nonzero probability. We show that a slight modification of the state has a threshold for singlet fraction, which cannot be exceeded anymore.