Libraries do it all the time - it's called weeding. Not every book can stay on the shelf until the end of time, and the ones that don't get checked out get pitched. Don't get the vapors, though. Large library systems, and those at the state and university level try to keep the last copy of a tattered book that has circulated well, especially if it's out of print.
As far as the British Library and the Shakespeare book goes, I can believe it. It's the British Library; they probably get 18th century volumes of Shakespeare's works in the mail every day, and it's not like it was the First Folio or anything.
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