Were there any saber-tooth tigers in the 1600s?

Basically, no.

The giant reptiles we usually mean when we say "Dinosaurs" lived from about 230 million years ago until about 65 million years ago. The only mammals alive in those times were very small, early mammals... And those only toward the end of that period.

The first saber toothed cats appeared around 40 million years ago, missing the main dinosaur party by roughly 25 million years.

However, as a sneaky fine-print addition, you could say that we are around in dinosaur times, since under modern phylogenetic taxonomy, all neornithes (modern birds) are classed as "dinosaurs".