Well, it IS the legal source. Now, there is no rule that I know of that says the suffix has to make sense - I could name my child Axlerod Fotheringay Humbolt VXIII and no one is going to check whether the appropriate number of A.F.H.'s existed prior, so a birth certificate record check won’t necessaryily help the OP.
Interestingly, though, the Social Security Administration does not use the suffix in assigning a SSN, only name, date of birth and parent names. I have no clue what they would do in the hypothetical case of twins born on the same day and given the same name with a different suffix - like John Jr. and John III.