Traditional text-based human-AI interactions often adhere to a strict turn-taking approach. In this research, we propose a novel approach that incorporates overlapping messages, mirroring natural human conversations. Through a formative study, we observed that even in text-based contexts, users instinctively engage in overlapping behaviors like"A: Today I went to-""B: yeah."To capitalize on these insights, we developed OverlapBot, a prototype chatbot where both AI and users can initiate overlapping. Our user study revealed that OverlapBot was perceived as more communicative and immersive than traditional turn-taking chatbot, fostering faster and more natural interactions. Our findings contribute to the understanding of design space for overlapping interactions. We also provide recommendations for implementing overlap-capable AI interactions to enhance the fluidity and engagement of text-based conversations.