There are many design techniques to support the co-design of tangible technologies. However, few of these design methods allow the involvement of users at scale and across diverse geographic locations. While popular in psychology, the story completion method (SCM) has only recently started to be adopted within the HCI community. We explore whether SCM can generate meaningful design insights from large, diverse study populations for the design of Tangible User Interfaces (TUIs). Based on the results of two questionnaire studies using SCM, we conclude that the method can be used to generate meaningful design insights. Drawing on a systematic review of 870 TUI papers, we then contextualise the strengths and weaknesses of SCM against commonly used design methods, before reflecting on our experience of using the method across two distinct domains. We discuss the advantages of the method (particularly in terms of the scale and diversity of participation) and the challenges (particularly around constructing meaningful story stems, and developing the correct level of scaffolding to support creativity). We conclude that SCM is particularly suitable to be used in the early stages of the design process to understand the socio-cultural context of deployment.