Translation:Does your sister have a big cat or not really?
huh....."или не очень?"? I put "Does your sister have a big cat or not so much?" and it took it!
I think that the question is asking whether the cat that the sister owns is actually big or not. Your answer doesn't convey that.
Translating this as "Is your sister's cat big, or not very" might get the idea across to an English speaker. The translation given would be more likely to be interpreted as asking whether that big cat I often see in your yard actually belongs to your sister, or is it just a stray.
How do you "not really" have a big cat? The answer is either yes or no. The word really is not required.