I have a few, but I'll start with this one as it's the funniest (and also the longest, sorry).
After DH and I married, we moved into an apartment with paper-thin walls. Our NDN, a college student with a penchant for partying, was pretty loud on a normal night (which had never bothered us). But when she woke us screaming bloody murder and banging on the walls one night at 3am, DH went around to check things were ok. We hadn't actually met her, but had seen her around and thought she looked normal enough.
Big mistake.
The first thing NDN did when she opened the door was scream something along the lines of "THANK GOD!", because apparently a spider had gotten in her apartment and that was the cause of all the blood-curdling screaming. She tried to pull DH (a huge, scary-looking man she'd never met) by the arm into her apartment in the middle of the night to "kill the spider for us." (Yes, "us". There were several other grown adults in the room, all somehow incapable of killing a spider.)
When DH politely declined hunting and killing a spider in a strange, drunken woman's apartment at 3am, and asked her about maybe keeping the screaming down just a bit instead, she got massively put out.
Now DH looks intimidating - he's basically a hairy giant with an enormous, dark beard - but he's a huge softie and absolutely hates confrontation, so he was incredibly nice and apologetic about it all. But drunken NDN got really angry and actually sent her BF to the door to "deal" with DH. BF was a meek, skinny and rather short guy who took one look at DH and obviously wanted no part of it. DH politely asked BF if they could maybe keep it down just a touch, and BF quite reasonably agreed. DH came back to bed.
Within minutes NDN is pounding repeatedly on our (shared) wall. HARD. We hear her shouting at the wall about how she's doing it on purpose to keep us up. DH sighs, goes back over, and says nicely that if they can't keep it down we'll have to call the police and we REALLY don't want to do that over a spider. NDN goes berserk, and threatens to call the police on us, shouting about how she KNOWS WHAT WE'RE TRYING TO DO, and she knows how "PEOPLE LIKE US" work because she's a PSYCH MAJOR. 
Finally, she calms down and agrees, and DH and I go to bed.
The next morning she knocks on our door looking completely and utterly hungover. She issues a slurred apology and introduces herself for the first time, while admitting she has absolutely no idea what she did/said last night but she "definitely doesn't want to be RUDE, since we're neighbors," etc. 