Obviously...

Each of the 4 small trapezoids have exactly one quarter of the area of the large trapezoid. Since the ratio of the areas of similar figures is equal to the square of the square of the ratio of the length or width of those figures, each side on the smaller trapezoid is half the length of the larger side.

Now consider this trapezoid:

It has side lengths $3, 7, \sqrt5$ and $\sqrt{13}$ . If we could stick 4 of these together into a big trapezoid, the resulting trapezoid would have side lengths $6, 14, 2\sqrt5$ and $2\sqrt{13}$ . We can see that the only way we could split up the sides of the larger trapezoid into the side lengths of the smaller trapezoids is by cutting them in half, like this:

This will not work, as the remaining area cannot be divided into 2 more congruent trapezoids. Therefore, this trapezoid is a counterexample, and not all trapezoids can be cut into four congruent smaller parts that are similar to the original trapezoid.