The guy who is overhearing the conversation with Woody Allen and Mira Sorvino at the horse racing ticket booths was playing piano in the next scene about the marriage anniversary.
Amanda tells Lenny that a day-old baby is available for adoption. Moments later, she tells him the baby was born that morning.
The scene at the school where Lenny and Amanda consider sending their child was shot at the City & Country School in New York, a world-famous progressive K-8 school in the Greenwich Village section of Manhattan. During their visit, a school administrator mentions that the child's test scores are acceptable. In real life, however, the school does not use standardized-test scores when considering prospective students.
The Belmont Park race caller announces that the last-place Eager Beaver is fading when Eager Beaver was actually gaining ground, and finishes the race a length-and-a-half behind the horse in front of him.
In the scene after Cassandra tells Lenny that his knee caps will be broken, the shadow of a boom mic is visible "following" him as he paces while talking on the phone to Amanda.