Usually? Are there examples where this isn't the case? Thanks for the explanations by the way.

No problem. If there are any examples where that isn't the case I'm not aware of them .

How about The Who Sell Out? I believe this qualifies as a concept album, but it does not tell a story.

I don't really consider The Who Sell Out to be a concept album. It has similar themes however. Tommy or Quadrophenia would be concept albums

BTW Townshend originally planned The Who Sell Out To be a concept album but it kind of changed as they got into it.

Dark side of the moon does not tell a specific story but I would call it a concept album.

I have a nomination that isn't prog-rock so I'm sure won't be that popular but I personally think it deserves a mention: Air - The Virgin Suicides
I don't know if any of you guys know this soundtrack album but I think it is great!

I forgot one: Alan Parsons Project - Tales of mystery and imagination. Its a great album.

Here are two conceptual musical adaptations of classic literature:

...Jeff Wayne's recording of H.G. Wells' War of the Worlds
...Rick Wakeman's working of Jules Vernes' Journey To The Center of The Earth

I love both of those albums, they could be the first two albums I ever got into, I was aboout 5 years old when I first got into them.