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Recently, prompt tuning (PT) has gained increasing attention as a parameter-efficient way of tuning pre-trained language models (PLMs). Despite extensively reducing the number of tunable parameters and achieving satisfying performance, PT is training-inefficient due to its slow convergence. To improve PT's training efficiency, we first make some novel observations about the prompt transferability of "partial PLMs", which are defined by compressing a PLM in depth or width. We observe that the soft prompts learned by different partial PLMs of various sizes are similar in the parameter space, implying that these soft prompts could potentially be transferred among partial PLMs. Inspired by these observations, we propose Fast Prompt Tuning (FPT), which starts by conducting PT using a small-scale partial PLM, and then progressively expands its depth and width until the full-model size. After each expansion, we recycle the previously learned soft prompts as initialization for the enlarged partial PLM and then proceed PT. We demonstrate the feasibility of FPT on 5 tasks and show that FPT could save over 30% training computations while achieving comparable performance.