I have to admit I was a little confused when I asked this question, as I thought that it was my Gmail where names and email addresses were connected.
But of course, it's not the local Gmail client that attaches a name to an email for display when receiving, it's whether or not the sender has a name attached to their email, as explained in the answer.
Okay... but, can I set Gmail up in a way so that it will replace a known email address with the name I have for them in my contacts in situations when someone sends me an email with no name set? Or, even better, over rides their name with the one I have set in any case (since I think of them by the name I have set, not what they have set)?
This is standard practise when it comes to phone numbers - when someone calls my Android phone, I see the name that I have set in my contact list. Hopefully this could be done with emails as well.