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Point cloud interpolation aims to recover intermediate frames for temporally smoothing a point cloud sequence. However, real-world challenges, such as uneven or large scene motions, cause existing methods to struggle with limited interpolation precision. To address this, we introduce DiffPCI, a novel diffusion interpolation model that formulates the frame interpolation task as a progressive denoising diffusion process. Training DiffPCI involves two key stages: a forward interpolation diffusion process and a reverse interpolation denoising process. In the forward process, the clean intermediate frame is progressively transformed into a noisy one through continuous Gaussian noise injection. The reverse process then focuses on training a denoiser to gradually refine this noisy frame back to the ground-truth frame. In particular, we derive a point cloud interpolation-specific variational lower bound as our optimization objective for denoiser training. Furthermore, to alleviate the interpolation error especially in highly dynamic scenes, we develop a novel full-scale, dual-branch denoiser that enables more comprehensive front-back frame information fusion for robust bi-directional interpolation. Extensive experiments demonstrate that DiffPCI significantly outperforms current state-of-the-art frame interpolation methods (e.g. 27% and 860% reduction in the Chamfer Distance and Earth Mover's Distance on Nuscenes).