Other mistake: When Alice asks Dan to make her some tea (after he tells her about him and Anna), he walks into the kitchen for just a moment, and comes back to find her gone. He also doesn't see her as he runs downstairs and looks outside. She couldn't have quietly gone down the stairs and out of the building in the short moment that Dan was in the kitchen. If she had moved quickly, there should have been a lot noise from her footsteps.
Other mistake: When Anna takes her first picture of Alice, she lifts the camera to her face, then the shot changes to faraway, and she lifts it to her face again.
Other mistake: At Anna's Exhibition, when Natalie Portman is smoking a cigarette, she's smoking a Red (Marlboro), but we can see the White Marlboro (Lights) box when she offers one to Larry (Clive Owen).
Answer: Anna says that Alice has "had quite a life" (after she read Dan's book that was based on Alice). Presumably Alice has been through difficult situations: abuse, neglect, etc. She considers herself a "nobody" and enjoys lying as a form of control. Which is why she never told Dan her real name, or everything about herself. She could never trust anybody completely. She also says "I'm the one who leaves." She prefers to leave men before they can leave her.