I'm going to go against the grain here and say it's not completely normal. and probably get flamed.
My first two DC slept through from about 15 weeks 7pm until 6ish am.
My third did not. She was never a very good sleeper, but by 5 months she was getting worse and worse. Until we were at a point of her waking every 30 mins in the night for about 3 weeks. DH and I were on our knees. At that age I do not believe in cc, it's not proven to work. But I remember saying to DH, we do something or we co-sleep.
It was always clear that she was not hungry every time, often when we tried a bottle she would pretty much refuse all but one a night.
We adopted the following approach.
*white noise
*naps always in the cot
*rubbing tummy to encourage her to sleep and to show her we were with her.
I'm not going to lie, she did not like it at first, but within a few days was getting much better and within 2 weeks was waking for 1 bottle a night.
I had numerous other friends with babies the same age as my youngest and her behavior was definitely not seen as normal by them, nor by our health visitor.
She went from being a baby who was regularly upset to a much happier baby. Looking back I think she was cronically exhausted like her parents.