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Animals can't talk, but they can think, so lines that look like dialog appear in thought bubbles. Sometimes animals can communicate with each other and reply to each other's thought balloons. This means that either the thought balloons are effectively translations of Animal Talk, or all animals are telepathic.

This is a comic trope - the film equivalent would be to sub-title an animal.

Examples:

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Newspaper Comics

Snoopy and Woodstock from Peanuts.
The animals from Garfield, apart from Odie.
Snowie from Tintin.
The animal cast of Footrot Flats.
Boot and other dogs from The Perishers, but not the crabs or insects.
In Marvin, this is how babies and animals communicate.
Animals in Mother Goose And Grimm use thought bubbles sometimes and Speech Bubbles other times.

Literature

Played with in Stanley Bagshaw and the Short-Sighted Football Trainer:
Grandma (thinks): "He's a boy of simple pleasures."
Cat (thinks): "Yes, that's just what I thought.
Cat (thinks): "Stone me: I'm a mind-reading cat!''

Webcomics

Used in Two Lumps. In fact, in this example, thought bubbles are used for conversation, while internal monologue is handled by Thought Captions.
Used by fish in Amazing Superpowers here.
Tally-Ho! implies (and has corroboration from Word Of God) that the animals do make sounds when they communicate, but the fox claims that a phone wouldn't transmit "dog thoughts." It does anyway.