Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$?
1. **Understanding the problem**: We need to find the minimum possible value of \( d \) such that every non-degenerate quadrilateral has at least two interior angles with measures less than \( d \) degrees.

2. **Sum of interior angles**: Recall that the sum of the interior angles of any quadrilateral is \( 360^\circ \).

3. **Extreme case consideration**: Consider a quadrilateral with angles \( 120^\circ, 120^\circ, 120^\circ - \epsilon, \epsilon \) where \( \epsilon \) is a very small positive number. This configuration ensures that the sum of the angles is \( 360^\circ \):
   \[
   120^\circ + 120^\circ + (120^\circ - \epsilon) + \epsilon = 360^\circ
   \]

4. **Analyzing the angles**: In this configuration:
   - Two angles are exactly \( 120^\circ \).
   - One angle is slightly less than \( 120^\circ \) (i.e., \( 120^\circ - \epsilon \)).
   - One angle is very small (i.e., \( \epsilon \)).

5. **Checking the condition**: For this quadrilateral, we have:
   - Two angles are less than \( 120^\circ \) (i.e., \( 120^\circ - \epsilon \) and \( \epsilon \)).

6. **Conclusion**: Since we can always find such a configuration where two angles are less than \( 120^\circ \), and no configuration can have fewer than two angles less than \( 120^\circ \) without violating the sum of \( 360^\circ \), the minimum possible value for \( d \) is \( 120^\circ \).

The final answer is \( \boxed{120} \).