You are right that one number only will have a Uniform distribution (or what you call "even"), however, it will be a discrete uniform distribution (if I understand correctly that the selected number is an integer) and therefore not a straight line parallel to the X-axis but a series of 16 dots parallel to the X-axis.
Also - I don't think that the sum of 3 uniform distribution is close enough to the bell curve (the Normal or Gaussian distribution). I am not sure how many you need, but probably at least 10.