Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation $w''\,{=}\,2w^{3}+zw+\alpha$ is of order at least $3/2$, and that any transcendental solution of the fourth Painlevé equation $2ww''\,{=}\,(w')^{2}+3w^{4}+8zw^{3}+4(z^{2}-\alpha )w^{2}+2\beta$ is of order at least $2$.