One possible problem can be a lack of fiber in the diet, especially in the induction phase of Atkins. This can cause constipation in some individuals, but can be corrected by supplemental fiber.
Another concern of the induction phase could be for those people who seem to be sensitive to cholesterol in foods.
Another problem is the withdrawal symptoms. Some people suffer greatly for a few days while they are adapting to a low carbohydrate lifestyle. I have known several people who felt they were unable to continue the diet and reverted back to high carb eating.
There are also concerns with possible mineral loss in bones and with kidney stones. Atkins refutes these claims, although supplemental calcium and other minerals may be a good idea for low carb dieters.

