According to the ad firm that came up with the tag, it's supposed to be a reference to Chevy's deep roots in American culture. OK, sure -- but if you need to explain it, it's not effective. The phrase is exceptionally bad, considering Chevy's rich history of iconic advertising. According to Jalopnik:
"This wouldn't be such a big deal if Chevy had never unveiled some of the most successful advertising lines ever. 'Like A Rock,' 'Heartbeat of America,' even the ancient 'See the USA in Your Chevrolet' -- all said something about why Chevys could be special and worth owning. Compared to those, 'Chevy Runs Deep' sounds less like a pitch and more like a diagnosis from a dermatologist.

It also reads like a phrase put together from a magnetic word-jumble on someone's refrigerator.