My friend's daughter has had the same symptoms at this time of year since she was just 2. This year (she's now 4) my friend is thinking for certain that it's hayfever, so I would say that they CAN get it so young.
As a sufferer myself, I'd say you'll know one way or the other in a couple of weeks or so - if the symptoms don't go, it could well be an allergy, not necessarily to pollen, but to anything.
I noticed that Boots now do an anti-histamine syrup suitable for age 2+, non-drowsy obviously. It's the same drug as I use, and I find it very effective - called loratadine, I'm not sure of the spelling!
Good luck