How can you tell if a bedroom set is made from pressed wood?

Close examination should reveal what parts are not made from real wood. If you find some, do not assume that the whole thing is made from man made materials.

If it is painted, this will be tougher. You could research or inquire of the manufacturer in addition to conducting your own examination. If the salesman is very knowledgable and honest, he will tell you what it is made of. For stained pieces, look for wood grain on the edges of the sides and top at the back of the piece. Remove drawers and inspect the interior (sides and under the top of the carcase) for grain. If there are any strong characteristics visible on the outside of any part (such as a burl or quilt pattern) it should be visible on the opposite side. A whorl or knot may travel from surface to surface and be visible on both sides, but not necessarily and certainly not as a mirror image. If you see such evidence, it will support whatever else you find which indicates it is wood. Look at what appears to be the laminations that make up a broad piece such as the top of a dresser. You will see straight borders along the length of the top where the grain is different on each side. If the third piece from an edge appears to be 4 inches wide, look at the underside and expect to see a 4 inch wide piece in the same spot. If not, you are likely looking at veneer, or worse, plastic/paper sheeting. Look at machined edges for grain. If there is none, it might be pressed product such as MDF or HDF. Edges could be banded with ornamental solid wood, in which case there will be a border as mentioned above.

Edges could also be veneer taped. It is unlikely that the grain of a solid piece runs perfectly parallel to the sides. Rather, it will cross at an angle and you should see it run across the edge. The grain on a taped edge will run in a different direction from what you would otherwise expect usually - along the length of the tape.

In my opinion, modern production furniture made entirely from real wood is very rare and expensive. If you are buying this from a store, I put my money on it using at least some man-made materials, even if it's artisan furniture such as Shaker or Mennonite. That is not necessarily a bad thing. Veneered plywood adds strength and stability (does not creep due to changes in humidity) and can be very expensive when veneered in woods such as cherry or walnut.