Hammy's quiet and heartbroken "I'm not stupid..." is pretty hard-hitting. Hasn't everyone had moments like that before, thinking that just because they're fun-loving and simple doesn't mean they're stupid.
The angrier tone of voice he has when he says it again while refusing to look Verne in the eye, implying that he's lost all respect for his friend, makes it a Wham Line.
"Still" may be one of the saddest songs Ben Folds has ever written.
The scene where the animals are all in crates after RJ takes the food and leaves. Their reaction is really what sells it.
Heather: I don't want to die, Dad. Not for real.
Ozzy: There there, sweetheart... we'll be okay.
When Ozzie thinks Heather is dead after she gets kicked by Gladys. Besides the fact that the poor guy thinks he just witnessed his only child get killed in a pretty brutal fashion (a less than ten pound possum getting kicked several feet into a wall is pretty brutal, even without any signs of trauma), which is sad enough, it becomes worse when you realize that Ozzie no doubt felt like he had lost everything, since his wife appears to have died before the start of the film, and now he thinks his daughter was killed as well.
It doesn't help that you get a quick glimpse, right before Heather 'wakes up', of Ozzie silently sobbing.