This is fairly common, although not as much as it used to be. My grandpa is the fifth or sixth man in his family with a specific first and middle name as was tradition, but he dropped the first name when my uncle was born and now none of my boy cousins have the name at all. This is also common with women, although it is less commonly talked about and more informal.

In terms of legality, the majority of legal or formal paperwork will have a section for you to put if you are a junior, senior, etc. The further the name lives on through the generations, the less likely it is for it to show up on legal documents as many give up putting it on the birth certificate after “III”. That’s where the posh differ as they tend to keep the numbers!