In my family, we have Bill Jones, his son Bill R Jones, and a grandson Bill G Jones.
Can the grandson properly use Bill Jones III (the third)?
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Not properly. The reason to use the numeral is to distinguish two people with the same name. So if they have different middle names, no need to use the numeral.
The Suffix (name) entry in Wikipedia notes:
According to The Emily Post Institute, an authority on etiquette, the term Jr. can be correctly used only if a male child's first, middle, and last names are identical to his father's (current) names. When a male child has the same name as his grandfather, uncle or male cousin, but not his father, he can use the II suffix, which is pronounced "the second".
The numeral is presumably not on any official documents (birth certificate, passport, etc.), so then adding a numeral (properly or improperly) presumably does not make it part of the legal name. Obviously this would depend on various legislations and regulations of the location of interest.
Finally, to tie this to genealogy. This adding numerals after names seems to very much be an American tradition. In the UK, the exception is the monarchy; just about nobody else uses numeral suffixes in the UK. There is some irony to Americans embracing this monarchical tradition, but that is neither here nor there. I do find it quite irksome when genealogists add numerals after their ancestors' names in family trees or other publications, when they did not use those suffixes at any time in their lives. Although it might be done with the best intention, to distinguish multiple people of the same name, it is misleading. It would be (in my opinion) much better to distinguish them in text in other ways, for example by appending a birth year such as "John Johnson (b. 1753)", not inventing a suffix like "John Johnson IV".