With DD we started cow's milk right at 12 months like you are "supposed to"
Within a couple weeks she had a constant "cold" and was misdiagnosed with a sinus infection
Elliminated cow's milk again, elliminated the problems.
Months passed and we decided to reintroduce cow's milk. So for the next couple years I got to keep buying whole milk which she may or may not drink (and no one else would). We would go in cycles of her drinking or not drinking. I decided to encourage it after two, though, because I was preganant and my supply plummetted. Now, at six, she drinks a mixture of cows, soy & almond milk (not at the same time, lol, just whatever she wants on a particular day).
With DS I skipped that and he just nursed until two. Around then he would start to drink a sip or two of cow's milk if he wanted. Now at 3.75 he generally drinks *some* cow's milk each day, but still under a cup. Many days under 1/2 cup.
No mammal actuallly *needs* milk after they have passed their natural weaning age (estimated 2.5-7 for humans) so milk intake is a personal preference. With DD being a rather picky vegetarian (very good grain & fruit consumption), I like her drinking milk for the protein. It is less important to me that DS drinks milk (esp since he is still nursing) because he has other protein sources.