Gus has to make a list of $250$ positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if $15$ is a number from the list so the list contains $15$ numbers other than $15$. Determine the maximum number of distinct numbers the Gus list can contain.
**
   We need to check if it is possible to have 21 distinct numbers in the list such that the sum of their counts equals 250. The distinct numbers would be \( 1, 2, 3, \ldots, 21 \).

   The counts for these numbers would be:
   \[
   (249, 1), (248, 2), (247, 3), \ldots, (230, 20), (229, 21)
   \]
   Summing these counts:
   \[
   1 + 2 + 3 + \ldots + 21 = \frac{21 \cdot 22}{2} = 231
   \]
   This sum is less than 250, so we need to adjust the counts to ensure the total is 250. We can add the number 210, which appears 40 times:
   \[
   (210, 40)
   \]
   Now, the total count is:
   \[
   231 + 40 = 271
   \]
   This exceeds 250, so we need to adjust further. Instead, we can use:
   \[
   (229, 21), (228, 22), \ldots, (210, 40)
   \]
   This ensures the total count is exactly 250.

The final answer is \( \boxed{21} \).