Iris Chang's "The Chinese in America" details many instances of this happening. Typically in regions where the Chinese population was very minor (for instance, in the Deep South), Chinese people were seen as curiosities rather than a particular race. She details several instances of conversations that more or less went like this:

Asian person: "Which bathroom do I use?"

White person: "Are you coloured?"

Asian person: "No, I don't think so."

White person: "Then use the white one."

On the other hand, sometimes the Chinese WERE classified as coloured. For instance, in 1924, Lum vs. Rice, the Supreme Court ruled that any non-white population could be judged as coloured.

In the end, it really depended on the local whites in the community. By the 1950s, after years of what basically amounted to Anglicization, the general trend was to allow Chinese students to attend white schools and use white facilities. Of course with Brown vs. Board of Education and the civil rights movement this became moot.