Reply to Remixing tracks quickly

Just been looking at videos on YouTube now.

Came across a few 2 Many DJ's interview videos where they said they made the Dizzee Rascal "Bonkers" remix after hearing it for the first time about 30 mintues before they were due to play. They made a version quickly and played it at that show then.

There was another one where they claim to have made a Daft Punk remix with Erol Alkan in a hotel room 10 minutes before they were due to play live where they then included it in the set.

Question 1: Would it even be possible to genuinely make a remix that quickly?

Question 2: If so, how?

Just wondering as some of their remixes are mind-blowing, like MGMT's Kids etc.

Only asking as at the moment I only mix/create using other peoples tunes and don't "remix" tunes as such. I add custom loops or beats created in Ableton to beef up some tunes, but I wouldn't call that a remix.