Like most English people of my generation I’m ignorant of grammar rules but in the reality of everyday usage this is one of those cases where you may see it written down but are very unlikely to hear it said.
The only time I have every heard anyone say ‘my partner and I’ is in a faux ‘posh’ English accent in reference to the Queen saying ‘My husband and I’ (I have no idea if the Queen ever said that but it is what people say when they impersonate her for some reason).
But written down it may be more natural to write ‘my partner and I’ than ‘me and my partner’, depending on the context of the sentence of course.