There have been a number of studies on sparse signal recovery from one-bit quantized measurements. Nevertheless, less attention has been paid to the choice of the quantization thresholds and its impact on the signal recovery performance. In this paper, we examine the problem of quantization in a general framework of one-bit compressed sensing with non-zero quantization thresholds. Our analysis shows that when the number of one-bit measurements is sufficiently large, with a high probability the sparse signal can be recovered with an error decaying linearly with the ?2-norm of the difference between the quantization thresholds and the original unquantized measurements. Specifically, by setting the thresholds sufficiently close to the original unquantized measurements, sparse signals can be recovered with an arbitrarily small error. By borrowing an idea from the Delta modulation, we propose an adaptive quantization scheme where the quantization thresholds are iteratively adjusted based on previous encoded bits such that they eventually oscillate around the original unquantized measurements with decreasing granular noise. Numerical results are provided to collaborate our theoretical results and to illustrate the effectiveness of the proposed scheme.