Question

What would happen if your pc boxes and party were full and you successfully caught a pokemon?

Answer 1

You should still be able to throw pokeballs but if it gets caught you will not be able to keep it.

The same thing happened in the first generation if a box got full - it didn't used to change boxes automatically. So if you caught a Pokemon you couldn't keep it.

Comments

But does it's data go to the pokedex

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That's for you to find out :)