Not officially, but the definition of a carry on bag is vague. Generally it is accepted to mean a larger piece of luggage, a sports bag, rollaboard or something similar. In addition to this you can usually carry a small purse or back bag, laptop bag or a similar personal item. Plus as mentioned they usually don't care about plastic shopping bags either.

If you try to bring two or three very large bags then sure, it will raise the alarm flags. Just be sensible.