No this is a myth. I have asked my Pediatrician as well as an ENT about this. Dairy should be avoided at times with stomach bugs because it can cause more irratation in the bowels. It does not increase mucus or make you stuffier.
For those that are legitimately allergic that could be a symptom of the allergy.
Unfortunately milk/diary has had bad rap over the past few years and now kids are lacking lot's of Vitamin D and are suffering the consequences of the anti-dairy campaigns. It may seem like it makes you stuffier just because it is more thick of a liquid then water. However it will not make any cold symptoms worse and kids still need lot's of it daily in one form or another.
Milk Myth #2: Drinking milk causes mucus.
Myth Buster: Milk consumption does not lead to mucus production or occurrence of asthma.
The belief has been held for years that milk causes mucus formation, although the few studies on this topic have failed to demonstrate any effect of milk on mucus production. Many people confuse the temporary, slight thickening of saliva after drinking milk with mucus. There is no scientific research showing that milk produces mucus in the airways or the throat. It will not worsen cold or asthma symptoms. In fact, drinking lots of fluids when you have a cold is important in speeding up recovery and may do your immune system some good.3
Dr. Dean: Way back in the 1920s, Arnold Ehret wrote a book called the Mucusless Diet System in which he put forth the idea that dairy products make mucus and cause colds. Well, this caught on big and has not let go. One reason for the tenacity is that drinking a big, ol' glass of milk does leave a mucusy feeling in the mouth. There is no truth to Ehret's theory though.
Mucus is produced de novo, by that I mean from mucus producing cells right in the bloodstream. Mucus has nothing to do with drinking milk fat or eating dairy products.
Ear, nose, and throat doctors were being asked about this so frequently, that about six years ago they decided to do a study to finally put it to rest. Researchers loaded people up with milk and cheese and measured mucus flow rates. They found that a dairy diet didn't make a lick of a difference.
Now, some people can't have dairy products because they are lactose intolerant, but that has nothing to do with mucus and colds. Lactose intolerance causes gas, cramps, and diarrhea.