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Despite the soaring use of convolutional neural networks (CNNs) in mobile applications, uniformly sustaining high-performance inference on mobile has been elusive due to the excessive computational demands of modern CNNs and the increasing diversity of deployed devices. A popular alternative comprises offloading CNN processing to powerful cloud-based servers. Nevertheless, by relying on the cloud to produce outputs, emerging mission-critical and high-mobility applications, such as drone obstacle avoidance or interactive applications, can suffer from the dynamic connectivity conditions and the uncertain availability of the cloud. In this paper, we propose SPINN, a distributed inference system that employs synergistic device-cloud computation together with a progressive inference method to deliver fast and robust CNN inference across diverse settings. The proposed system introduces a novel scheduler that co-optimises the early-exit policy and the CNN splitting at run time, in order to adapt to dynamic conditions and meet user-defined service-level requirements. Quantitative evaluation illustrates that SPINN outperforms its state-of-the-art collaborative inference counterparts by up to 2x in achieved throughput under varying network conditions, reduces the server cost by up to 6.8x and improves accuracy by 20.7% under latency constraints, while providing robust operation under uncertain connectivity conditions and significant energy savings compared to cloud-centric execution.