This study examines public organisations planning big data-driven transformations in their service provision. Without radical structural change or managerial system changes, leaders face dilemmas: simply bolting on big data makes little difference. This study is based on a qualitative empirical case study using data collected from the cities of Helsinki and Tampere in Finland. The three core dilemma pairs detected and connected to the big data-related organisational changes are: (1) repetitive continuity vs. visionary change, (2) risk-taking vs. security-seeking and (3) technology-based development vs. human-based development. This study suggests that organisational readiness involves not only capabilities; instead, readiness involves absorbing knowledge, making decisions, handling ambiguities, managing dilemmas. Thus, big data-related transformations in public organisations require embracing the world of dilemmas, since selected and cancelled experiments may each have valuable outcomes. The capability to act on intentions is a prerequisite for readiness; however, a preparedness to detect and address dilemmas is central to big data-related transformations. Thus, the ability to make dilemma decisions is a more complicated characteristic of readiness. In conclusion, our data analysis suggests that traditional public organisational and chance management approaches produce unsolved dilemmas in big data-related organisational changes.