In the first scene Kelly and Peg are looking at Peg's Valentines and Kelly is reading one card, when they switch cameras it's a totally different card in her hands.
The hotel bell boy that Bud steals the uniform from is taller than Bud but Bud's uniform fits perfectly.
On the Valentine's day card that Kelly forgot to pass onto Bud has no writing on either side so how did Peggy know it was for Bud? Kelly with her poor memory would not remember it was for Bud.