Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.
1. **Understanding the Problem:**
   We need to find the number of odd positive integers \( n \) less than 250 such that for every prime \( p \) dividing \( n \), \( p - 2 \) also divides \( n \). These integers are called twinning integers.

2. **Analyzing the Condition:**
   For \( n \) to be twinning, if \( p \) is a prime divisor of \( n \), then \( p - 2 \) must also divide \( n \). This implies that \( p - 2 \) must be a divisor of \( n \) for each prime \( p \) dividing \( n \).

3. **Considering Prime Divisors:**
   Let's consider the primes less than 250:
   - If \( p = 3 \), then \( p - 2 = 1 \), which trivially divides any \( n \).
   - If \( p = 5 \), then \( p - 2 = 3 \), so \( 3 \) must divide \( n \).
   - If \( p = 7 \), then \( p - 2 = 5 \), so \( 5 \) must divide \( n \).
   - If \( p = 11 \), then \( p - 2 = 9 \), so \( 9 \) must divide \( n \).

4. **Excluding Larger Primes:**
   - If \( p = 13 \), then \( p - 2 = 11 \), so \( 11 \) must divide \( n \). However, if \( 13 \) divides \( n \), then \( 11 \) must also divide \( n \), and \( 9 \) must divide \( n \). This would make \( n \) greater than \( 13 \times 11 \times 9 = 1287 \), which is greater than 250. Hence, \( 13 \) cannot be a prime divisor of \( n \).

5. **Possible Prime Divisors:**
   Therefore, the possible prime divisors of \( n \) are \( 3, 5, 7, \) and \( 11 \).

6. **Constructing Twinning Integers:**
   We need to consider combinations of these primes:
   - \( 3 \)
   - \( 5 \)
   - \( 7 \)
   - \( 11 \)
   - \( 3 \times 5 = 15 \)
   - \( 3 \times 7 = 21 \)
   - \( 3 \times 11 = 33 \)
   - \( 5 \times 7 = 35 \)
   - \( 5 \times 11 = 55 \)
   - \( 7 \times 11 = 77 \)
   - \( 3 \times 5 \times 7 = 105 \)
   - \( 3 \times 5 \times 11 = 165 \)
   - \( 3 \times 7 \times 11 = 231 \)
   - \( 5 \times 7 \times 11 = 385 \) (excluded as it is greater than 250)
   - \( 3 \times 5 \times 7 \times 11 = 1155 \) (excluded as it is greater than 250)

7. **Counting Valid Twinning Integers:**
   The valid twinning integers less than 250 are:
   \[
   3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231
   \]
   There are 13 such integers.

The final answer is \( \boxed{ 13 } \).