The semi-focal distance $c$ of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ is $c=4$.
To have two intersection points between the line and the hyperbola, the slope of one of the asymptotes of the hyperbola must be less than the slope of the line,
i.e., $\dfrac{b}{a} 2$,
and $a < c = 4$
Thus, the range of values for the length of the real semi-axis of this hyperbola is $(2,4)$
Therefore, the correct answer is $\boxed{A}$.
To have two intersection points between the line and the hyperbola, the slope of one of the asymptotes of the hyperbola must be less than the slope of the line, i.e., $\dfrac{b}{a} < 1$. By deriving the inequality relationship between $a$ and $b$, and then using $b= \sqrt{c^2 - a^2}$ to transform it into an inequality relationship between $a$ and $c$, a range for the eccentricity is obtained. Finally, considering that the eccentricity of the hyperbola is greater than $1$, a range for $e$ can be determined.
This problem mainly examines the simple properties of the hyperbola and the common characteristics of conic sections. When determining the range of values for the length of the real semi-axis of the hyperbola, note that it must be less than $4$.