For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$.
Find the largest possible value of $T_A$.
1. **Define the set and its properties:**
   Let \( A = \{x_1, x_2, x_3, x_4, x_5\} \) be a set of five distinct positive integers. Denote by \( S_A \) the sum of its elements:
   \[
   S_A = x_1 + x_2 + x_3 + x_4 + x_5
   \]
   Denote by \( T_A \) the number of triples \((i, j, k)\) with \(1 \le i < j < k \le 5\) for which \(x_i + x_j + x_k\) divides \(S_A\).

2. **Calculate the number of possible triples:**
   The number of ways to choose 3 elements out of 5 is given by the binomial coefficient:
   \[
   \binom{5}{3} = 10
   \]
   Therefore, there are 10 possible triples \((i, j, k)\).

3. **Example with specific values:**
   Consider the set \( A = \{1, 2, 3, 4, 494\} \):
   \[
   S_A = 1 + 2 + 3 + 4 + 494 = 504
   \]
   We need to check which triples \((i, j, k)\) satisfy the condition that \(x_i + x_j + x_k\) divides \(S_A\).

4. **Check each triple:**
   - For \((1, 2, 3)\):
     \[
     x_1 + x_2 + x_3 = 1 + 2 + 3 = 6 \quad \text{and} \quad 504 \div 6 = 84 \quad \text{(divides)}
     \]
   - For \((1, 2, 4)\):
     \[
     x_1 + x_2 + x_4 = 1 + 2 + 4 = 7 \quad \text{and} \quad 504 \div 7 = 72 \quad \text{(divides)}
     \]
   - For \((1, 2, 5)\):
     \[
     x_1 + x_2 + x_5 = 1 + 2 + 494 = 497 \quad \text{and} \quad 504 \div 497 \quad \text{(does not divide)}
     \]
   - For \((1, 3, 4)\):
     \[
     x_1 + x_3 + x_4 = 1 + 3 + 4 = 8 \quad \text{and} \quad 504 \div 8 = 63 \quad \text{(divides)}
     \]
   - For \((1, 3, 5)\):
     \[
     x_1 + x_3 + x_5 = 1 + 3 + 494 = 498 \quad \text{and} \quad 504 \div 498 \quad \text{(does not divide)}
     \]
   - For \((1, 4, 5)\):
     \[
     x_1 + x_4 + x_5 = 1 + 4 + 494 = 499 \quad \text{and} \quad 504 \div 499 \quad \text{(does not divide)}
     \]
   - For \((2, 3, 4)\):
     \[
     x_2 + x_3 + x_4 = 2 + 3 + 4 = 9 \quad \text{and} \quad 504 \div 9 = 56 \quad \text{(divides)}
     \]
   - For \((2, 3, 5)\):
     \[
     x_2 + x_3 + x_5 = 2 + 3 + 494 = 499 \quad \text{and} \quad 504 \div 499 \quad \text{(does not divide)}
     \]
   - For \((2, 4, 5)\):
     \[
     x_2 + x_4 + x_5 = 2 + 4 + 494 = 500 \quad \text{and} \quad 504 \div 500 \quad \text{(does not divide)}
     \]
   - For \((3, 4, 5)\):
     \[
     x_3 + x_4 + x_5 = 3 + 4 + 494 = 501 \quad \text{and} \quad 504 \div 501 \quad \text{(does not divide)}
     \]

5. **Count the valid triples:**
   The valid triples are \((1, 2, 3)\), \((1, 2, 4)\), \((1, 3, 4)\), and \((2, 3, 4)\). Thus, there are 4 valid triples.

Conclusion:
\[
\boxed{4}
\]