Learning from AI feedback (LAIF) is a popular paradigm for improving the instruction-following abilities of powerful pre-trained language models. LAIF first performs supervised fine-tuning (SFT) using demonstrations from a teacher model and then further fine-tunes the model with reinforcement learning (RL) or direct preference optimization (DPO), using feedback from a critic model. While recent popular open-source models have demonstrated substantial improvements in performance from the RL step, in this paper we question whether the complexity of this RL step is truly warranted for AI feedback. We show that the improvements of the RL step are virtually entirely due to the widespread practice of using a weaker teacher model (e.g. GPT-3.5) for SFT data collection than the critic (e.g., GPT-4) used for AI feedback generation. Specifically, we show that simple supervised fine-tuning with GPT-4 as the teacher outperforms existing LAIF pipelines. More generally, we find that the gains from LAIF vary substantially across base model families, test-time evaluation protocols, and critic models. Finally, we provide a mechanistic explanation for when SFT may outperform the full two-step LAIF pipeline as well as suggestions for making LAIF maximally useful in practice.