Yes, they do. Just consider the phenomenon of “multiple discovery.” Calculus, the polio vaccine, logarithms, the theory of evolution, the Higgs Boson model, light bulbs, the jet engine—each leap was made simultaneously by more than one researcher.
It is indeed a proven fact that great minds do think alike. In 1974, a study showed that 46% of research scientists made at least one discovery which led to a pretty convincing theory. It says that as people try to find solutions to particular problems, they tend to use the knowledge accumulated over time, which makes certain discoveries imminent.
Research scientists made at least one multiple discovery, leading sociologists to conclude that great ideas weren’t solely the result of great minds. (The idea that individuals change history is called the “Great Man Theory.” Leo Tolstoy hated the idea so much he wrote War and Peace to attack it.)
Rather, cultural conditions are in control. As people try to fix particular problems, they use a cultural knowledge that has accumulated over time, which makes certain discoveries almost inevitable. So if you’ve got a great idea, hurry up. You’re probably not alone.