No, bruxism is highly correlated with apnea. https://www.mayoclinic.org/diseases-conditions/bruxism/sympt....

I don't seem to be able to simultaneously breathe through my mouth and grind my teeth. They could still happen during different parts of the sleep cycle, though. Perhaps grinding during light sleep and mouth breathing during deep sleep?

What makes you wonder that?

Because with bruxism you typically have your mouth closed (I see no other way to grind your teeth).