Sure, the whole "I didn't cross the border; the border crossed me," thing gets overdone to make it seem like Mexicans were here all the time.
In fact cities like Los Angeles were predominantly Anglo-Saxon not so very long ago, but
1) border counties along the Rio Grande and elsewhere were predominantly Mexican and New Mexico was largely "Spanish" and Indian for a long time, and
2) large waves of Mexican immigration go back at least to the Mexican Revolution of a century ago.
That's about the time when many European immigrant groups came over.
There were a lot of “Mexicans”, especially in the Southwest, for a long time. I think it’s interesting that the number doubled between 1990 and 2000. Those are (not all, but many) the modern day Mexicans can came for the benefits and to “take their land back”.