[Question: Do Christians believe that cavemen existed?]

Answer:
No, I don't.

Read my answer to another guy's similar question:

Neanderthals were perfectly modern humans who lived during the first few hundred years after the Flood. Before the Flood, people were living to be over 900 years old, due to an atmosphere that protected them from the sun's damaging rays that cause aging.

After the Flood, this protective layer was gone, so the damaging rays came in and started wreaking havoc on our bodies. The people who lived after the Flood could still live for a few hundred years, but that began to dwindle from about 400 down to 300, then 200, and today, few people make it to 100. When a person reaches a great age, their facial bones start to change. Their skulls thicken, their brow ridges thicken, and their noses and ears enlarge. Many of the Neanderthals were found with rickets, which is caused by a Vitamin D deficiency. We get Vitamin D from the Sun. During the time after the Flood, there was a great amount of volcanic activity due to the plates that were shifting up and down and back and forth from the weight of the Flood water on the mantle. This caused a great amount of volcanic activity, which blocked out the sun, thus causing the Rickets.

After the Earth began to stabilize, the human race also began to stabilize, and since we couldn't live as long anymore, we ceased to have the bone-thickening symptoms that go with extreme old age. Neanderthals and modern Man are the same.