You can get some odd fractions by two parents having similar lineages. Like, if your mother is Irish, and your great-grandmother on your father's side is Irish, you would be five-eighths Irish. I'm having trouble finding a combination that gives you thirds, though.

It is a rounding and reduction of genetic markers.

21/64 Germanic markers equals 1/3 German in speech because everybody hates the twenty-one sixty-fourths German guy.

If you really want to get into the weeds, you get one half of your chromosomes from your mother, and one half from your father (most of the time, oh boy!), which should start the train rolling on the 1/2, 1/4, 1/8, 1/16.... BUT there is a chance for crossover events, where the chromosomes can, well, cross over each other and exchange parts of themselves. So your dad should be passing on 1/4 of your genes from his mother, and 1/4 of your genes from his father (and even that isn't 100% true, the only certain one would be if you're a male, you're going to get your grandfather's Y gene, you could get 23 of your grandmother's chromosomes and none from your grandfather), but he might pass on 52/106 of your grandmother's genes (not chromosomes, to those of you counting along at home... and I'm not saying you only have 106 genes, good lord) to you, and 54/106 of your grandfather's genes.

And that's just getting started on genetic funkiness.