What happens is that three of the steps in the 12-bit DAC result in the same value for all reads with the 10-bit AVR ADC.
The fourth step results in two values but this doesn't mean there is 1 LSB of noise. Often most of the readings are one of the values.
Here's an interesting test. How about you post me the two sets of data and I try to tell identify which is which ? 1024 points would be preferable.