Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities:
\[\left\{\begin{array}{cc}
\dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\
\dfrac{1}{6-z}+2\le x \\
\end{array}\right.\]
1. **Substitute variables**: Let \( z = 6 - a \), \( y = z - b \), and \( x = y - c \) where \( a, b, c > 0 \). This implies:
   \[
   y = 6 - a - b \quad \text{and} \quad x = 6 - a - b - c
   \]

2. **Rewrite inequalities**: The given system of inequalities is:
   \[
   \frac{1}{y - x} + \frac{1}{z - y} \leq 2
   \]
   \[
   \frac{1}{6 - z} + 2 \leq x
   \]

   Substituting the expressions for \( x \), \( y \), and \( z \):
   \[
   \frac{1}{(6 - a - b) - (6 - a - b - c)} + \frac{1}{(6 - a) - (6 - a - b)} \leq 2
   \]
   \[
   \frac{1}{6 - (6 - a)} + 2 \leq 6 - a - b - c
   \]

3. **Simplify the inequalities**:
   \[
   \frac{1}{c} + \frac{1}{b} \leq 2
   \]
   \[
   \frac{1}{a} + 2 \leq 6 - a - b - c
   \]

4. **Analyze the first inequality**:
   \[
   \frac{1}{c} + \frac{1}{b} \leq 2
   \]
   By the AM-HM inequality (Arithmetic Mean-Harmonic Mean inequality), we have:
   \[
   \frac{b + c}{2} \geq \frac{2bc}{b + c}
   \]
   Simplifying, we get:
   \[
   (b + c) \left( \frac{1}{b} + \frac{1}{c} \right) \geq 4
   \]
   Given \(\frac{1}{b} + \frac{1}{c} \leq 2\), we have:
   \[
   (b + c) \cdot 2 \geq 4 \implies b + c \geq 2
   \]

5. **Analyze the second inequality**:
   \[
   \frac{1}{a} + 2 \leq 6 - a - b - c
   \]
   Rearrange to:
   \[
   \frac{1}{a} + a + b + c \leq 4
   \]
   By the AM-HM inequality, we know:
   \[
   \frac{1}{a} + a \geq 2
   \]
   Therefore:
   \[
   2 + b + c \leq 4 \implies b + c \leq 2
   \]

6. **Combine results**: From steps 4 and 5, we have:
   \[
   b + c \geq 2 \quad \text{and} \quad b + c \leq 2
   \]
   Thus:
   \[
   b + c = 2
   \]

7. **Equality conditions**: Since \(\frac{1}{b} + \frac{1}{c} \leq 2\) and \(b + c = 2\), equality must hold in the AM-HM inequality, implying:
   \[
   b = c = 1
   \]

8. **Solve for \(a\)**: From \(\frac{1}{a} + a \geq 2\), equality holds when \(a = 1\).

9. **Determine \(x, y, z\)**:
   \[
   z = 6 - a = 6 - 1 = 5
   \]
   \[
   y = z - b = 5 - 1 = 4
   \]
   \[
   x = y - c = 4 - 1 = 3
   \]

Conclusion:
\[
(x, y, z) = (3, 4, 5)
\]

The final answer is \(\boxed{(3, 4, 5)}\).