For a great many reasons, some of which I sent to the poster, I am certain the label says TAMI, and not TARNI or TARMI, although I agree that it looks like that. I assume, Nicole that the hits you found are all contemporary sites from people who also read the label as Tarni? Or did you find an historic reference to Tarni?
The early references I found indicate that Tami Sportswear (and its subsidiary Tami, Inc) made skirts that were often paired with sweaters from Darlene Knitwear. I found no reference at all to a person named Tami Rich. Rather, in the 1960s through at least 1976, Tami sportswear/Tami Inc was owned by Athlone and newspaper articles mention the men who ran the company.