Sinatra was known as "old blue eyes," but his actor at first has brown eyes. In the next scenes, he indeed has blue eyes, either from processing or color contacts.
When the doctor is talking to Jack about all the medications he's on in episode 4, he says he's on gabapentin to relax the muscles. It is not a muscle relaxant and did not come to market until 1994. This episode takes place in 1961. He also says "the ritalin isn't helping you sleep either". It's a stimulant it's supposed to keep you awake, not help you sleep.
JFK's doctor mispronounces the drug Lomotil as LOM-oh-til instead of low-MOW-til.
Bobby's wife joins their kids in doing the Bunny Hop. This dance craze actually went through in 1952. They would hardly be doing it 9 years later.