G'day Sean, I am not a medical person but have spent many years
tracking these kinds of problems with little ones. What I repeatedly
found was that the child rarely had problems until ceasing
breastfeeding and wholly transferring to a formula based kind of
feeding. The coughing or sniffling problems usually established within
the four week period following, in some cases this was significantly
more profound when the mother also introduced other dairy based food
simultaneously.
In many cases moderate to severe eczema was also present.
Reflux involvment could also be a problem.
In virtually every case the baby formula was found to be cowsmilk based
even though the producers were averse to confirming this.
Switching immediately to a soy or other derived foodbase was often
helpful, providing it was quickly done.
In cases where the problem had substantially manifested, it involved
major ongoing problems with other compounding allergy affecteng
substances.
Immediate examination of all food items that the child is exposed to is
absolutely neccessary, you will likely find the cattle derivatives are
most of the problem.
Keeping the child sensibly cool at all times helps substantially.
It would be in your interest to read Professor Gary Huffnagle's
information regarding the adverse effects of antibiotics in these
cases. (google his name)
Where this problem is treated with antibiotics the children most often
develop asthma.
Cheers, Merlin.

I may be stoned for this, but what the heck.  If the onset of
your son's cough was a cold, you might want to check out
www.asthmastory.com.  He wouldn't be the first baby who was
diagnosed with asthma, which may actually be caused by bacteria
lingering from an illness.  I know two children (one of whom
started with a barking type of cough following illness at the age
of two) who have had their asthma resolved because it was caused
by bacteria, and finally (after several years) treated with
weekly doses of antibiotics for a longer than usual period (six
to twelve weeks).   My own asthma was resolved after 33 years,
when the bacteria was treated two + years ago.   Look at the
asthma research section, and also the files on the homepage, on
the lower left there for more information.  I'm the forum
moderator on that site - and NO - nobody is selling anything, and
it isn't a scam.