Is it true that if a people descend far enough down the line that they will no longer have genes in common...?

Actually it depends of how you define genetically. Scientist are able to trace a person's lineage for thousands of years. If two people were sibling in a tribe long ago, they will still have markers for that area. However, on a practicle level , they are unlikely to share traits such as eye color or baldness that we think of as genetic. They could also marry and have children as it is very unlikley they still carry similar recessive genes that could lead to brith defects.