Increasingly, organizations are pairing humans with AI systems to improve decision-making and reducing costs. Proponents of human-centered AI argue that team performance can even further improve when the AI model explains its recommendations. However, a careful analysis of existing literature reveals that prior studies observed improvements due to explanations only when the AI, alone, outperformed both the human and the best human-AI team. This raises an important question: can explanations lead to complementary performance, i.e., with accuracy higher than both the human and the AI working alone? We address this question by devising comprehensive studies on human-AI teaming, where participants solve a task with help from an AI system without explanations and from one with varying types of AI explanation support. We carefully controlled to ensure comparable human and AI accuracy across experiments on three NLP datasets (two for sentiment analysis and one for question answering). While we found complementary improvements from AI augmentation, they were not increased by state-of-the-art explanations compared to simpler strategies, such as displaying the AI's confidence. We show that explanations increase the chance that humans will accept the AI's recommendation regardless of whether the AI is correct. While this clarifies the gains in team performance from explanations in prior work, it poses new challenges for human-centered AI: how can we best design systems to produce complementary performance? Can we develop explanatory approaches that help humans decide whether and when to trust AI input?