What happens if I add corn before bringing the water to a boil?
What happens is not so different than when you do add at boil.  You just have less control over the perfectly cooked point.

At sea level, water boils at 100&deg;C and 212&deg;F.  However, when you add your room temperature (or from the fridge at 4&deg;C temperature), the water temperature drops.

How much you say? well it depends mostly on the weight of water and the weight of your vegetables.  If you add 1lb of vegetable from the fridge to 4 quart of boiling water, you'll end up with a mix that is maybe around 85&deg;C (185&deg;F).

At this point the stove tries to take the new mix back up to boiling point and your vegetables  continue to cook.

By adding the vegetables before boiling point, you'll reduced the mixed temperature even further.  

At any given temperature, different vegetables will take a certain amount of time to cook.  Since the temperature here is going up and down, knowing exactly when they're prefect gets much harder, that's why the recipes try to standardize on the boiling water at x minutes thing.  

That said, some food when taken through a range of temperature (say cold to hot) will have different components in them denature at different times and some things dissolving into the water before cooking begins.  This sequence might end up with unpleasant and unsightly things released into the water and the food not holding together well.

the bottom line:

If you were close to boiling when you added food, don't worry about it.  If you put the food in cold water then brought to boil, you're looking at a different recipe :)