I have this question that has been bugging me lately.

Is it possible to accidentaly generate the same adress twice on 2 different machines?
I mean, if you think of it, couldn't it be possible that i make a new adress on my pc to do a payment, and at the exact same time, someone else makes one too.
I know the chances are almost infinitely small, but in theory those 2 adresses could be the same right?  Does the client check with the network before making one? What about nodes that are disconnected but have empty adresses on them for later use?
How does the blockchain deal with that? Is there a safety protocol in place to prevent this? Would those 2 adresses have the same private key as well? Would the blockchain reject one of those adresses, and if so, which one?

I'm trying to understand as much as possible about bitcoin, so if anyone can enlighten me, that would be great!