This paper concentrates on cell-free massive multiple-input and multiple-output (MIMO) network with variable-resolution analog-to-digital converters (ADCs). In such architecture, all ADCs equipping at any access point (AP) can use arbitrary bit resolution to realize adaptive quantization and reduce power consumption. Under this circumstance, we first introduce a channel estimator based on linear minimum mean-square error (LMMSE) theory. On this basis, intra-AP and inter-AP bit allocation problems are investigated to maximize channel estimation quality subject to the total number of quantization bits. By leveraging on the statistical properties of the estimated channels and estimation errors, we then derive the theoretical expressions of the achievable uplink spectral efficiency (SE) for maximal ratio combining (MRC) and minimum mean-square error (MMSE) combining, respectively. Furthermore, to maximize the sum SE under the constraint of total ADC quantization bits, we also investigate intra-AP and inter-AP bit allocation problems for both single-user and multi-user scenarios. Finally, simulation results confirm that our theoretical analyses are correct and accurate. In addition, we resort to numerical results to achieve some new insights and verify the advantages and conclusions pertinent to the proposed bit allocation techniques.