Determine whether there exist a positive integer $n<10^9$, such that $n$ can be expressed as a sum of three squares of positive integers by more than $1000$ distinct ways?
1. **Define the problem and the goal**: We need to determine whether there exists a positive integer \( n < 10^9 \) that can be expressed as a sum of three squares of positive integers in more than 1000 distinct ways.

2. **Consider the form \( N = a^2 + b^2 + c^2 \)**: We are looking at numbers \( N \) that can be written as the sum of three squares of positive integers \( a, b, \) and \( c \).

3. **Range of \( a, b, \) and \( c \)**: Since \( n < 10^9 \), we need to consider the range of \( a, b, \) and \( c \). The maximum value for \( a, b, \) and \( c \) would be less than \( \sqrt{10^9} \approx 31622 \). However, for simplicity, we can use \( 10^4\sqrt{3} \approx 17320 \) as an upper bound.

4. **Count the number of possible triples \( (a, b, c) \)**: The number of possible triples \( (a, b, c) \) where \( 1 \leq a, b, c < 10^4\sqrt{3} \) is approximately:
   \[
   (10^4\sqrt{3})^3 = 3\sqrt{3} \cdot 10^{12}
   \]

5. **Count the number of possible values for \( N \)**: The number of possible values for \( N \) is at most \( 9 \cdot 10^8 \) since \( N < 10^9 \).

6. **Apply the Pigeonhole Principle (PHP)**: By the Pigeonhole Principle, if we have more triples than possible values for \( N \), then some \( N \) must be expressible in multiple ways. Specifically, we have:
   \[
   \frac{3\sqrt{3} \cdot 10^{12}}{9 \cdot 10^8} = \frac{3\sqrt{3} \cdot 10^4}{9} = \frac{\sqrt{3} \cdot 10^4}{3} = \sqrt{3} \cdot 10^3 \approx 1.732 \cdot 10^3 > 1000
   \]
   This shows that there must be at least one \( N \) that can be expressed as a sum of three squares in more than 1000 distinct ways.

7. **Improved estimate**: Using the improved estimate suggested, we have:
   \[
   10^{12}\left(\sqrt{\frac{10}{3}}\right)^3 > 6 \cdot 10^{12}
   \]
   With \( 10^9 \) numbers, the Pigeonhole Principle still works, ensuring that some number \( N \) has at least \( 1000 \) triples assigned to it.

Conclusion:
\[
\boxed{\text{Yes}}
\]