Depends on the plant. I'm no recycling expert, but you have to remember that recycling plants are a combination of computer and human labor.
Computers handle a lot of stuff easily: metals can be easily separated with magnets. Light plastics are separated with blowers, and some plastics can be detected with high-tech infrared / UV scanners.
So when I say "easy to recycle", I'm talking about Aluminum (non-magnetic but metal, so easy to sort automatically) or plastic bags (air-blowers catch these).
Everything else is dug around by hand by humans. If the humans can't figure it out, they throw it away immediately. Depending on the location, sometimes even Pizza boxes are thrown into the landfill by humans.
Pizza Box is one of those things that's highly localized. You can't put a Pizza Box into a paper-recycling plant: the oils contaminate the mix. But you can shred a Pizza Box into a compost bin.
IIRC, some areas only have paper-plants (so Pizza Boxes are thrown away there). So you gotta check with your local laws, regulations, and local plants to know for sure.
EDIT: It seems like some areas can't handle plastic bags. It really depends.
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For something like a K-Cup: it contains so little coffee that I find it unlikely that its worth the human's time to scoop out the coffee grounds. Its probably easier for the humans to throw a K-Cup away rather than manually labor in the recycling mix.