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Very, very possible. With my first, I thought I was nuts, and so did my mom. But it just never stopped. With my second, also at 4 months, I knew what it was and what it felt like. Some babies are pretty active, which is normal, and others are more mellow. Later on, [7 months for example] if you haven't felt him/her move at least a few times throughout the day, then you need to call your doctor. At that point, when he/she is so big [almost completely viable], no movement whatsoever could mean he/she's having problems.
It is quite normal for you to feel the baby kick or move. This will start in most pregnancies between 13-16 weeks. When you are between 20-30 weeks everyone will be able to feel the baby kick. My baby has started to kick where everyone can feel it and I am only 23 weeks right now. I hope this helped.