(H=human, C=chimp, G=Gorilla, O=orangutan)
Long long ago, in a laboratory far far away, scientists figured out that chimpanzees have 24 chromosomes in their sperms and eggs, whereas humans only have 23. Therefore, these great scientists theorized that two of our chromosomes might have fused together sometime in the recent past (aka million years ago.). Their theory made 3 predictions:
1) One of our chromosomes would look like two of the chimp chromosomes stuck together.
2) This same chromosome would have an extra sequence in it that looked like a centromere. Centromeres are the things in the middle that microtubules grab onto to divide a pair of chromosomes during mitosis.
3) It would also have telomeres (ends) but in the middle - and they would be in reverse order. Sort of like this:
See the "DNEEND" in the middle? That's what two telomeres would look like if two chromosomes were stuck together.
--From an excellent post by scigirl on IIDB.
As you might have guessed, all three predictions have been verified. While, as always, it's impossible to prove that an all-powerful being didn't create the evidence to trick us, the reasonable explanation is that humans and chimps share a common ancestor.