State, local and federal law enforcement often show up (sometimes simultaneously) at big Mob events that are known in advance, especially weddings and funerals. The FBI guys at Connie's wedding were a realistic touch.
The Apalachin meeting was a different affair. The NY State cops busted the meeting. Supposedly their curiousity was piqued when they saw a large number of Cadillacs approaching the home of Joe Barbara. Another version: local cops, who knew Barbara was a Mob guy, tipped the Staties because Barbara didn't pay them off.