Find the maximal possible value of $n$ such that there exist points $P_1,P_2,P_3,\ldots,P_n$ in the plane and real numbers $r_1,r_2,\ldots,r_n$ such that the distance between any two different points $P_i$ and $P_j$ is $r_i+r_j$.
1. **Claim**: The maximum value of \( n \) is 4.

2. **Construction for \( n = 4 \)**:
   - Let \( P_1, P_2, P_3 \) be the vertices of an equilateral triangle with side length 2.
   - Let \( P_4 \) be the barycenter (centroid) of this triangle.
   - Set \( r_1 = r_2 = r_3 = 1 \).
   - Calculate \( r_4 \) such that the distance between \( P_4 \) and any \( P_i \) is \( r_4 + 1 \). Since \( P_4 \) is the centroid, the distance from \( P_4 \) to any vertex \( P_i \) is \( \frac{2\sqrt{3}}{3} \). Thus, \( r_4 = \frac{2\sqrt{3}}{3} - 1 \).

3. **Case Analysis**:
   - **Case 1: All \( r_i \) are nonzero**:
     - Construct circles \( C_i \) centered at \( P_i \) with radius \( |r_i| \).
     - The hypothesis holds if:
       1. \( C_i \) and \( C_j \) are externally tangent if \( r_i, r_j > 0 \).
       2. \( C_i \) is internally tangent to \( C_j \) if \( r_i > 0 > r_j \).
     - If \( r_i, r_j < 0 \), then \( \overline{P_iP_j} \ge 0 > r_i + r_j \), which is a contradiction. Thus, at most one \( r_k \) can be negative.
     - If \( r_k \) is negative, \( C_k \) must be internally tangent to all other \( C_i \). If \( n > 2 \), there exist \( i \neq j \) such that \( r_i, r_j > 0 \), making \( C_i \) and \( C_j \) externally tangent, which contradicts \( C_k \) being internally tangent to both.
     - If no \( r_k \) is negative, all \( r_i > 0 \), and all \( C_i \) are pairwise externally tangent. For \( n \ge 3 \), adding \( C_4 \) requires placing \( P_4 \) inside \( \triangle P_1P_2P_3 \), which is uniquely possible, giving \( n = 4 \).

   - **Case 2: There exists \( r_k = 0 \)**:
     - At most one \( r_k \) can be nonpositive, otherwise \( \overline{P_kP_l} = r_k + r_l \le 0 \) for some \( k, l \), which is impossible.
     - All other \( r_i > 0 \), so \( P_k \) lies on \( C_i \) for all \( i \neq k \), and \( C_i, C_j \) are externally tangent for all \( i \neq j \neq k \).
     - All \( C_i, C_j \) are externally tangent at \( P_k \), which can only happen for at most two circles, giving \( n \le 3 \).

4. **Conclusion**:
   - The maximum \( n \) is 4, achieved by the construction with an equilateral triangle and its centroid.

The final answer is \( \boxed{ n = 4 } \).