Is the convolution between two CDF always well defined?

Given the integral convolution:

$$(F_X * G_X)(x)=\int_{-\infty}^x F_X(t)G_X(x-t)dt $$

and also that the CDF of random variables are bounded between the interval $(0,1)$ and the flipping of one of them in the product inside the integral make me think that:

$$\lim_{t\to\infty}F_X(t)G_X(x-t)=\lim_{t\to-\infty}F_X(t)G_X(x-t)=0 $$

Is always this integral well define, or there are cases where is not?