No, it cannot. You can incorporate an MIT licensed work into a work that is licensed differently, but compatibly (not a real problem with MIT), but now you have three works that are under different licenses: the MIT part, the "other" part and the whole. The MIT part stays MIT. Someone can cherry-pick just that file (or file snippet) and use it under MIT, without caring about the license of the whole.
What happens when a file was previously under MIT, the license changes, and a new change is made? Do I need to look at the git blame to find which parts I can use as MIT?
You don't need blame necessarily, just git log. You look at the latest version under your preferred license. That's the one you take your snippets from.
