Sorry if this question is long, I am trying to include all the details in case that helps.
My 2008 Honda Civic would not start after I had changed the brakes (left the trunk open and a door cracked so the interior lights were one). Every time I tried to start it turned over less and all signs pointed to it needing a jump start.
I used a 2002 Buick Century to jump start the car. The issue with the Century was that its battery it didn't have clear indication of positive or negative (Turns out is had a slight outline for positive and negative on the top that thin dust hid, please don't be like me and actually check) so I assumed the bright red label near one post and a black label near the other signified the positive and negative.
Needless to say my assumption was wrong. The batteries began to short and melted the protective plastic layer on the jumper cables (If this happens to you, make sure you only touch the clamps and avoid the cables, because while preventing potential further damage is good, second degree burns are no fun).
One I had the cables off and treated my burns I went to check the warning lights of each car to see if I broke anything. To my relief the Century started fine with no warning lights, and to my amazement so did the Civic.
Why did the failed jump start get the Civic to start? The battery didn't have enough charge to start the car, and hooking the jumper cables up backwards should have only drained the battery further.
Update: Ultimately the issue was the starter. If the car had been started recently (within the last few hours), then it wouldn't work. The time spent getting the other car moved over and attempting a jump-start allowed the starter to cool off enough. Replacing the starter resolved this issue. (Thankfully this didn't stump me for over 3 years, I just forgot to update the post)