Ask Mr. Smarty Plants

The terms 'native' and 'invasive' aren't mutually exclusive.  Even though the majority of the plants on the Invasive and Exotic Plants list of the Center for Invasive Species and Ecosystem Health are, indeed, non-native exotics—such as Ligustrum japonicum (Japanese honeysuckle) and Melia azedarach (chinaberry)—there are a number of natives on the list as well.  For example, the native Toxicodendron radicans (Eastern poison ivy) is on the list and certainly almost everyone considers it an invasive, unwelcome plant.  The surprising ones are ones that we consider attractive such as Hamelia patens (Firebush), a native North American plant—native to Florida, Puerto Rico and the Virgin Islands.  You can see a distribution map on the USDA Plants Database.  Surprisingly, the source responsible for it being on the list is Florida Keys Invasive Species Task Force.  Another attractive native on the list is Iris missouriensis (Rocky mountain iris) and the source for it is the California Noxious Weeds list. 

I don't know the exact reasons that Hamelia patens and Iris missouriensis are considered invasive by the two listing agencies.  Some plants are considered invasive because they interfere with agricultural crops; but I suspect for these two it is probably because they are growing and spreading prolifically to the detriment of other native plants in the areas.

The short answer to your question, however, is that, for whatever reason, a native plant can be listed as invasive or noxious along with non-native exotics.