The outline of Gondwana, a once-mighty continent connecting Australia, Africa, Antarctica, and South America, was based in large part on the distribution of a genus of ancient tongue fern called Glossopteris, which was found in all the right places. However, much later Glossopteris was also discovered in parts of the world that had no known connection to Gondwana. This troubling discrepancy was—and continues to be—mostly ignored. Similarly a Triassic reptile called Lystrosaurus has been found from Antarctica all the way to Asia, supporting the idea of a former connection between those continents, but it has never turned up in South America or Australia, which are believed to have been part of the same continent at the same time.
There are also many surface features that tectonics can't explain. Take Denver. It is, as everyone knows, a mile high, but that rise is comparatively recent. When dinosaurs roamed the Earth, Denver was part of an ocean bottom, many thousands of feet lower. Yet the rocks on which Denver sits are not fractured or deformed in the way they would be if Denver had been pushed up by colliding plates, and anyway Denver was too far from the plate edges to be susceptible to their actions. It would be as if you pushed against the edge of a rug hoping to raise a ruck at the opposite end. Mysteriously and over millions of years, it appears that Denver has been rising, like baking bread. So, too, has much of southern Africa; a portion of it a thousand miles across has risen nearly a mile in 100 million years without any known associated tectonic activity.