In the pilot episode of The New Adventure of Old Christine, Christine enrolls her son Ritchie in Westbridge, an exclusive private school in West Los Angeles. How did she get the underachieving Ritchie into such an elite school? She lied and said he was 1/16th Native American.
For a TV show, this seems like a reasonable bit. Not only does it fit Christine's personality, but it could be true. That is, the pale-skinned Ritchie character could be 1/16th Indian and it wouldn't be obvious.
Moreover, I imagine an elite school like Westbridge would be concerned about its ethnic diversity. I imagine it might accept Christine's claim without checking it. So the bit works for several reasons.
Nor is the issue of race forgotten after that. Christine's black friend Barb calls Westbridge "Whitebread." In the sixth episode (The Other F Word), Christine becomes obsessed with how white and blond the student body is. She campaigns to bring in a black student--until the black parents say they left the previous school because there were too many "fags."
As with the other mention of Indians on Old Christine, the creators are using Natives without really thinking about them. But this mention works better than the subsequent one did.
For more on the subject, see TV Shows Featuring Indians.