When Connie is talking with the homeless man at the train station, he hands her a sanitary wipe and as she wipes her hands her nails are painted bright red (no chipping)and manicured nicely. This seems a stretch as she has been living on the streets for nine months, with no money for enough food let alone nail polish.
When Connie sits down to a meal of soup eaten according to proper etiquette, the bowls of soup are tilted AWAY from the diners. Then the older Mrs. Winterbourne (Shirely MacLaine) is offered her medication by the chauffeur who takes away her wine glass. She says, ungrammatically, "Who is working for who?" It should be "whom".
In the scene where Bill goes into Connie's room to find her packing, he buttons his blazer before sitting down to make a pro/con list. Later, while still making the list, his blazer is unbuttoned. (1 hour, 4 minutes into the movie)
While eating lunch, in successive shots Connie and Bill move from Boston Common to Paul Revere's House to the Public Garden, then back to Revere's House; while the Garden and the Common are adjacent, Revere's house is much farther away from them than implied.
When Connie asks where the train is going, she is told that it is going to Boston. The passenger car exteriors have stripes of purple and gold. The only passenger service that goes from New York to Boston is AMTRAK and their colors are red, white and blue so it cannot be a Boston bound train.
The goof item below may give away important plot points.
Grace alters her will to include Connie and the baby, but since Grace thinks Connie is Patricia, Connie is identified in the Will as Patricia, hence, Grace's bequest would not go to Connie or the baby.
As the priest calls Connie "Patricia" at the wedding, we can assume that she is named on the marriage license as "Patricia Winterbourne", hence, she and Bill would have to be married under her legal name in order for their marriage to be valid.