The close-ups when the president walks Sally to her door after their date, were shot two months apart. They had to stop shooting because of weather, but you can see that it's raining on one shot, but not the other.
When Grace comes to the former president to tell him of her feelings, the shot looking at him (from her perspective) shows it to be almost sunset with a very orange sky. When looking back to her (from his perspective) the sky is very blue.
When Handy puts the ring on Sally's finger, his hands are still quite dirty. In the next shot, and also after she pushes him to the ground his hands are completely clean.
When the president talks to his staff in his office right before the elections you can see that his light blue collar is flexed on both ends.
In the next shot it is plain and braced looking.
When Ex-President Monroe Cole drops Sally off after their date, he gets out of the left side of the vehicle. As a protectee of the US Secret Service, he would be required to sit on the right side of the vehicle behind the agent in the front passenger seat.
Charlotte Cole is the ex-wife of former President Monroe Cole. According to current protective procedures of the US Secret Service, once a former First Lady divorces a former President, she no longer receives US Secret Service Protection.
The newborn Labrador puppies eyes are open and are far too big to be newborns. In real life, the mother would not have allowed Sally and her assistant to handle the pups or wipe off their afterbirth; she licks the afterbirth off the pups during delivery.
It is mentioned in a newscast at the beginning of the movie that Monroe Cole was a two term President. Since he succeeded real life President Bill Clinton (as mentioned by one of his employees) who served from 1993-2001, his second term would have ended in 2009, not 2004 when the movie takes place.
The President points out a meteor to Sally as he is walking her to her door. However it is raining (and thus cloudy) and would be impossible to see.
When Handy and the President are playing golf for Sally, the President says "I'll take a seven", referring to how many strokes it took him to get the ball in the hole. Handy replies "actually, it's 15" but his lips are clearly saying "14". This happens again, in a close up, after Handy describes each of the president's strokes.
Sally brings a box of Handy's stuff over to him when she's breaking up with him (her friend is driving). As she gets in the truck to leave and pulls the door closed, a crewmember holding a bounce card is reflected in the door.
When the President picks up a knife to study in the hardware store, reflections of the camera man and crew are visible.
(around 1 hr. and 1 min.) The camera crane and cameraman's shadow is visible on the road, after Handy gets shoved away.
"Hail to the Chief," played at the reception for Monroe Cole, should never be played for anyone except the current president and is inappropriate in that situation. Correct protocol dictates that "The Star-Spangled Banner" should be played as the proper salutation for former presidents.