A recent example, I was getting married to Sylgja, my character is a Stormcloak but had only taken Whiterun and Ingrod Ravencrone came to congratulate me. She's happy for both of us but eventually the army I am part of will see her exiled.
Another time, on my way to join the Dawnguard, this character hadn't fought any vampires, none for the companions and just headed straight to the fort when the guards started talking about it, no random attacks. First thing I get asked as I enter Dayspring Canyon "You've probably killed lot's of vampires haven't you." I always try to get Laid to Rest completed early now so that that assumption will be true.
Then theres the classic Cidnha Mine moment everyone gets, "Here's your Shiv, promise you won't ever use it on me..." A few minutes later, "Go take care of Grisvar the Unlucky then we can leave this mine for good."
What awkward in game moments can you remember?