For what positive integers $n \geq 3$ does there exist a polygon with $n$ vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?
To determine for which positive integers \( n \geq 3 \) there exists a polygon with \( n \) vertices (not necessarily convex) such that each of its sides is parallel to another one of its sides, we will analyze the problem step by step.

1. **Case \( n = 3 \):**
   - A triangle has 3 sides. For each side to be parallel to another, we would need at least 4 sides (since parallelism is a pairwise property). Therefore, it is impossible for a triangle to have each of its sides parallel to another side.
   - Conclusion: For \( n = 3 \), it is impossible.

2. **Case \( n = 4 \):**
   - A quadrilateral can have opposite sides parallel, forming a parallelogram. This satisfies the condition that each side is parallel to another side.
   - Conclusion: For \( n = 4 \), it is possible.

3. **Case \( n = 5 \):**
   - Suppose a pentagon satisfies the property. Then there would have to be a set of three sides all parallel to each other. However, in a pentagon, any three sides must include at least two adjacent sides. Two adjacent sides cannot be parallel in a simple polygon.
   - Conclusion: For \( n = 5 \), it is impossible.

4. **Case \( n = 6 \):**
   - A regular hexagon has opposite sides parallel, satisfying the condition.
   - Conclusion: For \( n = 6 \), it is possible.

5. **General Case for Even \( n \geq 4 \):**
   - For any even \( n \), a regular \( n \)-gon (polygon with all sides and angles equal) will have opposite sides parallel. This satisfies the condition.
   - Conclusion: For even \( n \geq 4 \), it is possible.

6. **General Case for Odd \( n \geq 7 \):**
   - Consider a regular \( (n-3) \)-gon and attach a square to one of its sides. This construction ensures that each side of the resulting polygon is parallel to another side. The regular \( (n-3) \)-gon has \( n-3 \) sides, and the square adds 4 sides, making a total of \( (n-3) + 4 = n+1 \) sides. Adjusting this construction to ensure \( n \) sides can be done by appropriately choosing the attachment point and ensuring the sides are parallel.
   - Conclusion: For odd \( n \geq 7 \), it is possible.

The final answer is \( \boxed{ n } \) is even or \( n \geq 7 \).