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We study an adaptive transmission scheme based on variable-rate turbo bit-interleaved coded modulation (VR- Turbo-BICM). The proposed coding scheme employs punctured turbo codes. A continuously varying transmission rate can be obtained by changing the code rate through both puncturing of the coded bits and adapting of the modulation constellation size. The main results are elaborated in two parts. First, we derive a closed-form expression for a set of achievable rate bounds (called rate thresholds) for VR-Turbo-BICM by employing recent results on the parallel channel performance of turbo code ensembles and the BICM parallel channel analysis model. The derived rate threshold is expressed as a fraction of the capacity of BICM with Gray mapping, where this fraction is a turbo code weight spectrum parameter. Simulation results illustrate that introduced rate thresholds predict well the rate versus SNR performance of VR-Turbo-BICM for a wide range of codeword error probabilities and codeword lengths. Next, based on a simplified rate threshold, we derive a power, puncturing rate, and modulation constellation size assignment policy for a slow fading channel.