The current version of Adobe Digital Editions cannot read books purchased from B&N (that are DRM protected). Note that if you have ADE installed, it may register itself with your computer's OS for opening '.epub' files - so it may certainly try to open them.
Also as others have pointed out, you do not need ADE to get things on your Nook. Adobe DRM has been 'augmented' so that storefronts have a choice about whether to offer 'classic' DRM (requiring registration with Adobe), or 'new' DRM (DRM key uses a 'ID/password'; in the case of B&N, they use your credit card number as the password). To date, B&N is the only storefront that uses the 'new' DRM, as far as I know. However, the current version Adobe Reader Mobile SDK (used by reader software) supports both, and presumably many non-Nook readers that have come out in the last year or so are using that as well (with the notable exception of Sony). For example Pandigital reader, and Bluefire for iOS can read B&N ebooks.
Adobe had announced ADE would be updated in late 2010, but it looks like that date has slipped.
There is a free B&N desktop application you can use if you want to read B&N books on your computer.
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