Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.
1. **Given Information and Setup:**
   - We have a cyclic quadrilateral \(ABCD\) with \(BC = CD = 2\).
   - \(I\) is the incenter of \(\triangle ABD\).
   - \(AI = 2\).

2. **Using the Incenter-Excenter Lemma:**
   - The Incenter-Excenter Lemma states that if \(I\) is the incenter of \(\triangle ABD\), then \(I\) lies on the angle bisector of \(\angle A\) and \(\angle D\).
   - Since \(AI = 2\), we can use this information to find relationships between the sides and angles of the triangle.

3. **Analyzing the Cyclic Quadrilateral:**
   - Since \(ABCD\) is cyclic, the opposite angles sum to \(180^\circ\).
   - Let \(F\) be the intersection of \(AE\) and \(BD\). We need to find the minimum value of \(BD\).

4. **Using Similar Triangles:**
   - By the properties of cyclic quadrilaterals and the given lengths, we can set up similar triangles.
   - Let \(BF = x\) and \(DF = y\).
   - By similarity, we have the ratio \(\frac{3}{x} = \frac{y}{1}\), which simplifies to \(xy = 3\).

5. **Applying the AM-GM Inequality:**
   - By the Arithmetic Mean-Geometric Mean Inequality (AM-GM), we have:
     \[
     \frac{x + y}{2} \geq \sqrt{xy}
     \]
   - Substituting \(xy = 3\), we get:
     \[
     \frac{x + y}{2} \geq \sqrt{3}
     \]
   - Therefore, \(x + y \geq 2\sqrt{3}\).

6. **Conclusion:**
   - The minimum value of \(BD\) is achieved when \(x + y = 2\sqrt{3}\).

\(\boxed{2\sqrt{3}}\)