Well, actually, Cleveland mob assosciate Carmen P. Zagaria refused the offer of his sponsers, Capo Joseph C. Gallo & soldier Thomas "The China-Man" Sinito, to be formally inducted, preferring instead to continue running his drug operation as he had been (although he was paying tribute to them anyway for protection, so i dont know why he'd supposedly refuse. THe guy later became a rat, so maybe the story is dubious.)

Gambino assosciate Salvatore Ruggiero, the brother of Gotti side-kick Angelo "Fat-Ange/Quack-QUack" Ruggiero, seemed content to work his heroin rackets on the fringes rather than follow his brother and maternal uncle Aniello Dellacroce into Cosa Nostra.

Andrew Gigante has served time for Union infiltration and extortion despite his father Chin Gigante's apparant redusal to formalyy induct him.

I dont think anybody who gets as far as the ceremony would be stupid enough to pike out at that point. ("Take this card a repeat after me. And let me prick your finger. Ill kiss you after" "Hmm, actually fellas this might not be for me.") Indeed.