What is this Danish Cabinet/Shelf for?  

Handy-lookin' cabinet.
I assume that the bottom piece was intended for folded linens, that sort of thing. It'd be great for shirts, T-shirts, scarves, accessories, rock collections, whatever. Me likey.
Perhaps it was offered in modular pieces- original buyer could mix & match to suit their specific intended use.
Obviously, the two would also be useful split apart; but I can't imagine the room where the bottom cabinet would be useful yet the top shelving was superfluous.
I wouldn't feel obligated to store the same TYPE of item in top half as bottom, which I think is what your question implies(?).

Isn't that interesting . . .
I've not seen a bottom-mounted tambour door before. I wonder if it is counterbalanced so that it closes easily.
Is there a shadow of the upper cabinet on the top of the "bureau" -- indicating that the two have always been together ? Your repurposing of the pieces seems quite logical. Open shelves like these are perhaps the most common type of case built, because they have the widest possible use. Bookshelves are one obvious function . . .

My guess
would be that the bottom unit would have been used to store cutlery and table linens while the top to display china. I wonder if the top component might have hung on the wall slightly above the bottom? Either way I like that bottom component.

If the top were intended specifically for china,
there'd usually be a shallow channel running along the back length of the shelves, to display plates standing.

usually
good to know. do you have a better guess?

Top reads as a bookshelf to me.
I'd use the top for books, and maybe the bottom for my collection of vintage magazines & pamphlets.
But, one could come up with a thousand uses for that great bottom cabinet.

Maybe it came from a departme...
Maybe it came from a department store? There is something commercial looking about it.

Thank you all!
It's interesting to see some actually like the bottom piece. For me, this is not very functional. I store T-shirts in those drawers, but as you can see the drawers are very shallow and if I put more than 2 shirts, the top layer would be too high...
They may well be not together to begin with, as I don't see any signs that indicate as such. The wood/color match perfectly and it's obvious that they were from the same company/brand, but as william-holden-caulfield mentioned these could be parts of a modular system. I didn't think about that!
SDR, tambour door is not counterbalanced, so unless you lock it it will come down. It looks pretty but is not very useful so I always leave it open.
China cabinet? I don't know. I don't see any hung or channel as some mentioned. I do see, however, small holes in rows on the shelves that can hold the pins or wires. Perhaps I'm missing vertical shelves/dividers?
I think I'll continue useing them as separte pieces. The top part has short hairpin legs now and looks pretty cute. I may convert the bottom part to a TV stand, storing a DVD player in the bottom.