Two sides of a triangle measure 4 cm and 9 cm. Determine which of the following choices cannot be the measure of the third side?

Let $a$ and $b$ be the lengths of the two given sides of the triangle. By the Triangle Inequality, the third side, $c$, must satisfy the inequalities $|a-b| < c < a+b$. Thus, $|4-9| = 5$ is the only value that cannot be the measure of the third side of the triangle. The answer is thus $\boxed{5}$.