Abstract: Reasoning about conjuncts in conjunctive sentences is important for a deeper understanding of conjunctions in English and also how their usages and semantics differ from conjunctive and disjunctive boolean logic. Existing NLI stress tests do not consider non-boolean usages of conjunctions and use templates for testing such model knowledge. Hence, we introduce ConjNLI, a challenge stress-test for natural language inference over conjunctive sentences, where the premise differs from the hypothesis by conjuncts removed, added, or replaced. These sentences contain single and multiple instances of coordinating conjunctions ("and", "or", "but", "nor") with quantifiers, negations, and requiring diverse boolean and non-boolean inferences over conjuncts. We find that large-scale pre-trained language models like RoBERTa do not understand conjunctive semantics well and resort to shallow heuristics to make inferences over such sentences. As some initial solutions, we first present an iterative adversarial fine-tuning method that uses synthetically created training data based on boolean and non-boolean heuristics. We also propose a direct model advancement by making RoBERTa aware of predicate semantic roles. While we observe some performance gains, ConjNLI is still challenging for current methods, thus encouraging interesting future work for better understanding of conjunctions.