Strictly speaking the backup must be based on what you actually have bought (i.e. if you strip the DRM from your purchase and store that, that's fine). That means you cannot download essentially an identical copy, that's still naughty. I know this is idiotic, but apparently whoever writes laws doesn't know this.
There are also some other subtleties. E.g. I recently stumbled across the fact that in Germany apparently complete books (as opposed to excerpts) are specifically exempted from the right for private copies unless
- you either transcribe them completely manually, or else
- the book has been out of print for at least two years.
This law probably stems from the widespread introduction of photocopiers, but now that e-books are a thing, too, it has the somewhat absurd consequence that whereas it's perfectly legal to copy music I have purchased across my various devices and their backups, technically I'm not allowed to do the same thing for e-books (i.e. for example keeping an e-book both on my computer and my phone and in their respective backups, too).
