Can you be 1/12th Cherokee? I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "1/16th", or generally $1/2^n, n \in \mathbb{N}$.

Obviously you could also be some summation of same to achieve 3/32nds, etc., but will an irreducible fraction with the numerator 1 always need a power-of-two denominator? More generally does it always require a power-of-two denominator?

Assumptions (as per comments):


Nobody can trace their lineage infinitely
Incest is OK, including transgenerational, but someone can't be their own parent.
This depends on the model. Instead of arguing that we have only $46$ chromosomes and cross-overs or whatever the mechanism is called are not that common, let us assume a continuous model.
That is, a priori, everybody can be $\alpha$ Cherokee for any $\alpha\in[0,1]$ and the rules are as follows


Everybody has exactly two parents. 
If the parents have Cherokee coefficients $\alpha_m, \alpha_f$, then the child has $\alpha=\frac{\alpha_m+\alpha_f}2$
In a sufficiently large but finite number of generations ago, people had $\alpha\in\{0,1\}$


It follows by induction, that $\alpha$ can always be expressed as $\alpha=\frac{k}{2^n}$ with $k,n\in\mathbb N_0$ and $0\le k\le 2^n$. Consequently $\alpha=\frac1{12}$ is not possible exactly (though for example $\frac{85}{1024}\approx\frac1{12}$ would be possible).
It doesn't matter if there is any type of inbreeding taking place anywhere in the tree (or then not-tree) of ancestors.
The only way to obtain $\alpha$ not of this form would involve time-travel and genealogical paradoxes: If you travel back in time and paradoxically become your own grandparent and one of the other three grandparents is $\frac14$-Cherokee and the others are $0$-Cherokee, you end up as a solution to $\alpha=\frac{\frac14+\alpha}4$, i.e. $\alpha=\frac1{12}$.
There is more than one way to count genetics. Suppose that 120 years ago all four pairs of your mother's great grand parents (8 in all) were pregnant with their first child, and both pairs of your father's grandparents (4 in all) were pregnant with their first child. 

Thus at that moment in time you have 12 ancestors, of mixed generations back, just entering child-bearing years. 

If exactly one of those twelve ancestors is full-blooded Cherokee, and the other eleven have no Cherokee blood, then:

Yes, you are clearly 1/12th Cherokee.

Whether you are counting back in even generations or in even years is an arbitrary choice, and it is incorrect to state that only one makes sense. Perhaps there are other valid ways to count ancestry back that would enable other fractions and counting.