I have seen many jokes about Applejack lying, :scoots: Rarity being selfish, and Rainbow Dash not being loyal. -_- I get a good laugh out of these jokes but it made me wonder what the elements mean. A theory I have is that they are not related to what the character is or how they act. Applejack tells lies a lot, Rarity does things for her own gain, Rainbow Dash leaves in hard times, Pinkie Pie gets depressed, Fluttershy acts rudely, and Twilight is not always very magical or a good friend.
Instead of the elements representing what the characters are, I believe they represent what the characters value and strive for. Sometimes the things they value the most are also the things they struggle with the most just like us. If their characters were defined by the elements they may be more 2-dimensional and less approachable.
Extra Note:
During the premiere of season 2, Discord reverses each character's element. Applejack, Rarity, and Rainbow Dash end up being the easiest to convert. That seems to correlate with them struggling with their values. Fluttershy and Pinkie Pie, however, live more like their elements which makes them harder to convert (especially Fluttershy).