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The event Thom refers to in this song might very well have occurred during Radiohead's June 3rd-7th 2003 stay in New York City. According to this page there was heavy rain during this period, particularly on june 4th and 7th:
JUNE
4 - 2.75" of rain fell, a record for the date. Shortly before the rain ended 0.66" fell in the hour between 8-9PM. The high temperature was just 57, nineteen degrees below average.

7 - Nearly eight inches of rain fell since May 21, with rain falling on thirteen of these eighteen days. And every day but one had cooler than average temperatures. Today's rainfall totalled 0.90" and fell largely between 11AM-7PM. Five more inches of rain would fall in June, which became the rainiest on record with 10.26" measured.
An early typed up draft for the lyrics, taken from the pdf files of theeraser.net:



Transcript
Q: "What have you learned about yourself - as a songwriter - from making The Eraser?"

Thom: "I got a lot more confidence. I go through phases where I have absolutely no faith in anything I've done at all. But I was actually talking about what I was doing again. I'd ring up a friend, say 'Listen to this', and play him the bass riff on 'And It Rained All Night'. It was things like that, little pockets of excitement that I'd missed for so long."
Thom: "'And It Rained All Night' has this enormously shredded-up element of 'The Gloaming', not that you'd ever [notice]. I remember doing that in New York. I couldn't sleep one night, and it was one of those New York things, where the rain just chucks down. The rain was so loud."
Thom: "When we stopped, I started listening really heavily to all that Fela Kuti stuff and that whole idea of just staying on one chord and letting the groove go really impressed me. 'Black Swan' on The Eraser is like that. Also 'And It Rained All Night' which really was a homage to the Talking Heads' Remain In Light and Byrne/Eno's My Life In The Bush Of Ghosts. The bass line is pure Bush Of Ghosts. I even use talking on the track."
Thom: "my favourite was 'and it rained all night', just because i'd never written a lyric like that before. it was basically a cut-and-paste of something i'd written, where i had my lounge just covered in bits of paper, and one was four pages long, which i cut down and cut down--all the way through thinking "this is never going to work". then we actually ended up recording it on a full moon through the night, because i have one of those big, fat telescopes my partner bought me, and since nigel is the only one who knows how to use it, when he comes to my house it's like 'come on, set it up for me'. so i'd go up to the roof and look at the moon an then run back downstairs and quickly write away. back and forth, it was really good, actually; it surprised me to write that lyric. and it reallly surprised me that i got nigel's voice in the headphones at the end going, 'yeah, that's good', because all the way through i was thinking 'this is so wack, it's never going to work'."
Several lines, that are taken from those early lyrics above, appeared as pairs of two lines each in the 'scrapbook' section of radiohead.com since 2004. They are arranged here in the same sequence as in the text above:







These are the finished handwritten lyrics, taken from theeraser.net:

In 2006 Burial remixed 'And It Rained All Night'. This version was released as a download on december 17th 2007 before appearing on a series of three 12" vinyl remix EPs in january 2008. Later that year the Eraser remixes were also released on CD in Japan:
It took Thom three years following the release of The Eraser to put together a proper tour based on the album. Recruiting Nigel Godrich, Flea, Mauro Refosco and Joey Waronker to form the band, the group played a first set of shows in october 2009 and another in april 2010, initially under the name "?????", since Thom had not yet named the project "Atoms For Peace". The tour ended with a solitary gig in august 2010 in Japan. Each night, the main set would consist of the The Eraser played in its entirety:
The song had its live debut on october 2nd 2009 at the Echoplex in Los Angeles at the first gig of Thom's then unnamed band, consisting of himself, Flea, Nigel Godrich, Joey Waronker and Mauro Refosco: