No, but, can you imagine?
The Ninth Doctor meets Rose, with whom he forms a platonic, non-romantic friendship, and she helps him recover from his post-Time War traumas. Eventually, however, Rose is out of his life. But he doesn’t mope around, because he meets badass Doctor-to-be Martha Jones, his equal in strength and badassery and adventurous spirit, and they becomes a badass adventurous duo. It’s Martha who witnesses Nine regenerating into the Tenth Doctor. It’s weird for a while, because she felt like she lost her badass buddy, and feels like he’s too erratic for her taste. Ten ends up becoming too full of himself and something of a megalomaniac, and Martha decides she has had enough, it’s not fun anymore, and she want out. Ten then realises what he has become, and that his behaviour caused him to lose another dear friend. Finally, he meets Donna. And she fills the void Martha left inside him. Donna, like Martha, is his equal, and their relation follows more or less in the same lines, except with Donna it’s more crazy-fun.
Sometimes I wish this is how it should have gone.