Cats and kittens can get colds just as we do. In many cases, they have been exposed to something before arriving at the shelter and the stress of change can bring on a cold. (Most kitties are not big fans of change.)
It’s never fun to have a cold, but just like with us, a cold will usually pass for a kitty within a few days. Occasionally, cats or kittens will develop secondary complications from a cold that will require prompt treatment. See the list of symptoms on the Foster Resource Page to help you decide what to do.