One of the important conjectures in set theory in the 60s was that strongly inaccessible cardinals are measurable. This was disproved by Tarski in 1960. Soon after this Erdős and Hajnal realized that they were very close to Tarski's result in their paper "On the structure of set mappings" which appeared in 1958.
Another miss was in the paper of Erdős, Hajnal, and Milner ("On sets of almost disjoint subsets of a set") from 1968 where they went pretty close to discovering Silver's famous theorem on GCH at singular cardinals of uncountable cofinality.