Yes, Dag, but there are literally hundreds of thousands of Mexican-American dual nationals.
I know several.
Every person born in Mexico is Mexican by birth (if they have a US parent, they're also Americans).
Every person born outside Mexico from a Mexican parent is Mexican by birth (they can apply for a passport with no trouble): and many Mexican-Americans do.
Every person who is granted the Mexican citizenship by naturalization can keep their other citizenship (though they have to swear that they'll side with Mexico in an eventual -but of course highly improbable- war against their home country).
So Zippo's question falls by itself.