What happens if we train a new Large Language Model (LLM) using data that are at least partially generated by other LLMs? The explosive success of LLMs means that a substantial amount of content online will be generated by LLMs rather than humans, which will inevitably enter the training datasets of next-generation LLMs. We evaluate the implications of such "regurgitative training" on LLM performance. Through fine-tuning GPT-3.5 with data generated either by itself or by other LLMs in a machine translation task, we find strong evidence that regurgitative training clearly handicaps the performance of LLMs. The same performance loss of regurgitative training is observed on transformer models that we train from scratch. We find suggestive evidence that the performance disadvantage of regurgitative training can be attributed to at least two mechanisms: (1) higher error rates and (2) lower lexical diversity in LLM-generated data as compared to real data. Based on these mechanisms, we propose and evaluate three different strategies to mitigate the performance loss of regurgitative training. First, we devise data-driven metrics to gauge the quality of each LLM-generated data instance, and then carry out an ordered training process where high-quality data are added before low-quality ones. Second, we combine data generated by multiple different LLMs (as an attempt to increase lexical diversity). Third, we train an AI detection classifier to differentiate between LLM- and human-generated data, and include LLM-generated data in the order of resemblance to human-generated data. All three strategies can improve the performance of regurgitative training to some extent but are not always able to fully close the gap from training with real data. Our results highlight the value of real, human-generated data in training LLMs, which cannot be easily substituted by synthetic, LLM-generated data.