I think your question involves the portion control and caloric intake. You could theoretically be eating less but taking more calories through liquids and not being aware of what you are ingesting.
The band is a tool to help you limit your portions. It is easy to cheat if you want to by continuing to consume dense, high calorie foods that lack nutrition and spread it through the day.
If you are not inclined to graze or snack, stay away from milkshakes, ice cream, and sugary drinks/snacks, and increase your activity levels you should lose weight.
For me I have problems with portion control when I am hungry. I am a structured eater, 3 meals a day, and I rarely snack.