Species are defined by breeding that produces fertile offspring. As you pointed out, merely being able to breed is too broad a critereon.
The number of chromosomes is not necessarily an indicator of closeness between species. Humans are born with extra or missing chromosomes all the time, but it doesn't make them any less human (although it may render them sterile). There are plants that have 2, 3, 4 or more copies of the same chromosomes.
The discovery of DNA only provides even more evidence that humans are related. In fact, research has found that humans actually have a much shallower genepool then most animals. You share more DNA with a person of a different race that has the same height then you do with a same race person of a different height.