LLM cascades are based on the idea that processing all queries with the largest and most expensive LLMs is inefficient. Instead, cascades deploy small LLMs to answer the majority of queries, limiting the use of large and expensive LLMs to only the most difficult queries. This approach can significantly reduce costs without impacting performance. However, risk-sensitive domains such as finance or medicine place an additional premium on avoiding model errors. Recognizing that even the most expensive models may make mistakes, applications in these domains benefit from allowing LLM systems to completely abstain from answering a query when the chance of making a mistake is significant. However, giving a cascade the ability to abstain poses an immediate design question for LLM cascades: should abstention only be allowed at the final model or also at earlier models? Since the error patterns of small and large models are correlated, the latter strategy may further reduce inference costs by letting inexpensive models anticipate abstention decisions by expensive models, thereby obviating the need to run the expensive models. We investigate the benefits of "early abstention" in LLM cascades and find that it reduces the overall test loss by 2.2% on average across six benchmarks (GSM8K, MedMCQA, MMLU, TriviaQA, TruthfulQA, and XSum). These gains result from a more effective use of abstention, which trades a 4.1% average increase in the overall abstention rate for a 13.0% reduction in cost and a 5.0% reduction in error rate. Our findings demonstrate that it is possible to leverage correlations between the error patterns of different language models to drive performance improvements for LLM systems with abstention.