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The rings of power (or maybe just the One Ring) prolongs life but does not increase it. This allowed Gollum to live way beyond a normal lifespan which is explained in the lore. However, he loses the Ring and it is found by Bilbo in the goblin caves during The Hobbit. Bilbo is fifty years old when he finds the Ring and gives it to Frodo when he is 111 but he is described in the books to be very well preserved for his age.
I'm not sure how much time passed between Bilbo giving the ring to Frodo and Frodo arriving at Rivendell the first time but I assume it to be within a few months to a few years. In that short time after Bilbo gives the ring to Frodo, old age starts manifest itself and both visibly and physically his age starts to catch up with him. Now Bilbo is still within a natural life span but without the Ring he becomes much closer to the frail hobbit his age should be.
My question is, if time caught up to Bilbo so quickly after losing the Ring, how was Gollum able to live 60+ years without the Ring without his true age catching back up to him? I don't recall in the books but in the movies he is still pretty limber for someone who is a few hundred years old.