My youngest son was the same, since his first hospital admission aged 10 days. Long story short he had silent reflux, which he was aspirating causing lung infections. This was made worse when I started solids which included cows milk (he was breastfed) but it was only fairly recently we discovered a cows milk protein allergy. It's made an amazing difference to his health discovering this, and conincidentally he came off cows milk at the same time he started daycare and has not been constantly sick the way everyone told me he would be.

Anyway my point is, if you suspect cows milk take him off it. And by this I mean butter, cheese, cream in cooking, everything. If it doesn't make a difference then I would recommend pursing allergy testing to see if you can find out why he is so sick, because it is not normal for a child to be constantly sick no matter what the experts tell you. I was told the same thing numerous times and it was only at my insistence that my son had his first chest x-ray, which showed pneumonia! Apparently the fact he was happy indicated he was well, when in fact he was happy because he is a happy kid and he didn't know that feeling unwell was a reason to be unhappy because he was unwell from birth