"Broken up" may not be the best choice of words, but you did not explain which synonym would make you happier: "divided", "segmented", "fragmented", "split", etc.

The DNA of a nucleated (a.k.a. eukaryote) cell is indeed split into multiple chromosomes and the number of the chromosomes and the number of genes on each chromosomes and the sequence of the genes on each chromosome are normally constant for a species and very similar for closely related species. The cells that have an incorrect number of chromosomes (which happens when a cell division does not work correctly) will normally die soon, because their DNA is incomplete.

Only when a cell has all the normal chromosomes, but also some extra chromosomes, it has a chance to survive, even if the extra chromosomes may interfere with some internal processes. Because superfluous chromosomes are much less harmful than missing chromosomes, there have been cases, rare at animals, but frequent at plants, when the entire genome has been doubled, when some cell division has failed, but the descendants of that cell have survived.

There are animals for which the number of the chromosomes and the sequence of the genes on them has remained unchanged for many hundreds of millions of years, though there are also animals where the DNA has been completely rearranged, because some chromosomes have fused into a single chromosome, other chromosomes have split into multiple chromosomes, and on some chromosomes the genes have been shuffled.

Nowadays, it is known that it is very likely that the common ancestor of all animals except comb jellies had 29 chromosome pairs (humans have 23 pairs). In most animal branches there were more chromosome fusions than chromosome splittings, so in the present most animals have fewer than 29 chromosome pairs.

Among the animals with the most conservative genomes are some sponges, some jellyfish, many echinoderms and some of their relatives, i.e. acorn worms, the lancelets, some nemertean worms and some bivalves.