Hey Nick, I finally noticed the one-eighth Cherokee.
Part Cherokee in The Mindy Project
"Navajo" in Happy Endings
1/16 Cherokee in Modern Family
The idea seems to be to give plain white characters some ethnic "flavor." It might be offensive to say they're part black or Asian, but anyone can claim to be part Native.
By doing so, the shows can pat themselves on the back for mentioning Indians while doing nothing to explore modern Indian life. In particular, they don't have to address the vast gulf between the characters' privileged existence and reservation-based poverty.