I'd be surprised if that was the case.

<UPDATED>
< Well surprised I am ! further research reveals that Harry is alledged to be an old Middle English form of Henry>

More likely its a result of transcription - Harry and Henry look very similar in written records.

Next suggestion is that 'arry and 'enry probably sound similar and so may have been incorrectly recorded.

Other common confusions - Jones, James, and Janes ( Jaynes) - some may recall Jane James - male !

Watkin, Gwatkin, Walkin, Watkins,.......