I'd say: Both dreams of Jim Bishop and his mother came true.
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Ye Thu(3) Both Jim Bishop's dream and his mother's dream came true.This is the best one.
khoff Ye Thu(3) Both Jim Bishop's dream and his mother's dream came true.This is the best one.(1) suggests that Jim Bishop came true (which makes no sense), and his mother's dreams came true.(2) technically means the same as (3) but is more confusing.(3) is the most clear.(4) suggests that Jim and his mother each had two dreams that came true, and so does this suggestion:
Ye ThuWhy can't "the" be used in front of "dreams"?One determiner can be used in a noun phrase, but not more than one. (There are exceptions, of course.).