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Hickson Compact Groups (HCGs) are dense configurations of 4 to 10 galaxies, whose HI (neutral gas) 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. The HI deficiency of HCGs is expected to increase as the HI morphological phase progresses along this sequence, potentially making it a useful proxy for evolutionary phase. We test this hypothesis for the first time with a large sample of 38 HCGs with VLA HI observations that are uniformly reduced and analysed with a purpose-built pipeline. However, we find little evidence that HI deficiency can be used as a proxy for the evolutionary phase of a HCG in either of the first two phases, with the distribution of HI deficiency being consistent in both, although it does greatly increase in the third phase. This appears to be the result to three factors: a) there is already a broad range of HI deficiencies in Phase 1 HCGs, possibly due to their differing locations relative to large scale structures; b) the timescale for major interactions and morphological changes is, in general, considerably shorter than the timescale for the destruction or consumption of HI gas; and c) some groups have their HI content rejuvenated by the late addition of a new gas-rich member (for which we added a new sub-phase, 3c, to the established evolutionary sequence). Finally, across all HCGs studied, we identify only a few cases where there is strong evidence for the existence of a previously proposed diffuse HI component in the intra-group medium, which might be detectable with improved observations. This work was completed with considerable attention paid to scientific reproducibility, and all reduction and analysis has been made public via Github and Zenodo. (Abridged)