Hyperparameter optimization (HPO) and neural architecture search (NAS) are powerful in attaining state-of-the-art machine learning models, with Bayesian optimization (BO) standing out as a mainstream method. Extending BO into the multi-fidelity setting has been an emerging research topic, but faces the challenge of determining an appropriate fidelity for each hyperparameter configuration to fit the surrogate model. To tackle the challenge, we propose a multi-fidelity BO method named FastBO, which adaptively decides the fidelity for each configuration and efficiently offers strong performance. The advantages are achieved based on the novel concepts of efficient point and saturation point for each configuration.We also show that our adaptive fidelity identification strategy provides a way to extend any single-fidelity method to the multi-fidelity setting, highlighting its generality and applicability.