The fat cat ran at the rat.

[curiousmarcus]

The fat cat ran at the rat.

It's from a book, Nat's Cat.

Is this natural? I would tend to say ​The fat cat ran after the rat.

 

[curiousmarcus]

Could running at somebody always be understood as with an intent to attack? The sentences I got from Fraze.it seem like it.

 

[Skrej]

As Piscean mentioned, the purpose of the sentence is focused more on phonics and alphabetics skills, rather than grammar.

One early step of learning to read is using rhyme and alliteration to increase phonemic awareness and decoding skills, beginning with monosyllabic words for blending practice and recognizing word families (in this sentence, the '-at' word family).

Multi-syllabic words require a bit more competency with phonemic awareness to differentiate between the syllable division rules, so that's likely why they didn't use 'after' - plus it's a different word family.