So let's say you are a druid or ranger and have an animal companion. Over time, your animal companion receives bonus Hit Dice. This might have a number of consequences: they gain feats and skill points for progressing this way, for example. However, would you rule that these are indeed BONUS Hit Dice and that they aren't ACTUALLY gaining Hit Dice? Because when a creature gains Hit Dice, for every four they gain, they raise one of their ability scores by 1 just like PCs.
If you do think the animal is actually progressing in Hit Dice, would you allow someone to raise his or her animal companion's Intelligence score above 2? This would technically make the creature sentient and allow it to learn a language (although still be unable to speak it unless it had the necessary physiology). However, one of the defining characteristics of an animal is that its Intelligence is only 1 or 2 (herbivores often have the former while predators usually claim the latter). Is it really an animal anymore? If not, is it a viable candidate for an animal companion? If it ceases to be an animal companion, does it retroactively lose all its Bonus HD and become just a regular creature again? If so, why would you allow it to advance to Intelligence 3 in the first place?