When playing Fallout 4, what happens if you get all the perks? Do you get an achievement or are you just not able to get any more perks?
Nothing happens when you get all perks, and there is no perk-related achievement either.
If you are playing on PC, you can easily test this by using the following console command:
and then purchasing all the perks.
The game still allows you to go beyond level 272, and still awards you with perk points (and will nag you about redeeming them through your Pip-boy). You just won't have any more perks to buy, unless you have a mod that adds more of them.
Nuka World and Far Harbor each add a new perk for each SPECIAL, for a total of 14 additional perks. Redeeming them all still doesn't change anything, other than make you even more powerful than you already are.