Context. Hickson compact groups (HCGs) are dense configurations of four to ten galaxies, whose H I morphology appears to follow an evolutionary sequence of three phases, with gas initially confined to galaxies, then significant amounts spread throughout the intra-group medium, and finally with almost no gas remaining in the galaxies themselves. It has also been suggested that several groups may harbour a diffuse H I component that is resolved out by interferometric observations. Aims. The H I deficiency of HCGs is expected to increase as the H I morphological phase progresses along the evolutionary sequence. If this is the case, H I deficiency would be a rough proxy for the age and evolutionary state of a HCG. We aim to test this hypothesis for the first time using a large sample of HCGs and to investigate the evidence for diffuse H I in HCGs. Methods. We performed a uniform reduction of all publicly available VLA H I observations (38 HCGs) with a purpose-built pipeline that also maximises the reproducibility of this study. The resulting H I data cubes were then analysed with the latest software tools to perform a manual separation of emission features into those belonging to galaxies and those extending into the intra-group medium. We thereby classified the H I morphological phase of each group as well as quantified their H I deficiency compared to galaxies in isolation. Results. We find little evidence that H I deficiency can be used as a proxy for the evolutionary phase of a compact group in either of the first two phases, with the distribution of H I deficiency being consistent in both. However, for the final phase, the distribution clearly shifts to high H I deficiencies, with more than 90% of the expected H I content typically missing. Across all HCGs studied, we identify a few cases where there is strong evidence for a diffuse gas component in the intra-group medium, which might be detectable with improved observations. We also classify a new sub-phase where groups contain a lone H I-bearing galaxy, but are otherwise devoid of gas. Conclusions. The new morphological phase we have identified is likely the result of an evolved, gas-poor group acquiring a new, gas-rich member. The large spread of H I deficiencies in the first two morphological phases suggests that there is a broad range of initial H I content in HCGs, which is perhaps influenced by large-scale environment, and that the timescale for morphological changes is, in general, considerably shorter than the timescale for the destruction or consumption of neutral gas in these systems.