The voiceover states that this time period was "before Apalachin." Now Mafia buffs know what Apalachin is, so for them there's no problem. But others watching this scene wouldn't know what Henry was talking about. Maybe the meaning of Apalachin could have been clarified in some way. But that's a very minor flaw, not something that significantly diminishes the movie.
He also said, "...and before Crazy Joe started a war..." which was ca. 1960-61.
I think Scorsese was trying to establish that Henry was "with" the Mob even as a teenager. Perhaps lost on a lot of viewers but a bit of authentication.
