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So you find what looks like a cardboard container full of yummy ice cream or you see that your favourite coffee shop does paper cups. You remember something about waxed paper. Hooray.
STOP
To make paper or cardboard water proof, they are laminated with polyethylene, a plastic resin. These products are in effect very thin plastic containers reinforced.

Other Issues
cardboard containers are made from virgin wood because there are major problems using recycled paper. Regulations are strict about what materials you can use to package food and drink and recycled paper isn’t strong enough.

Recycling
Because these cups are made from paper and plastic they are difficult to recycle. The parts have to separated. Though this can be done it is a complex procedure which adds to the cost of the recycled product.
many recyclers say that they don’t recycle paper cups. Though some claim to. It’s a murkey scenario at best.

Compostable Alternatives
There are compostable cardboard products for food on the market. They are lined with a clear, certified-compostable, cornstarch plastic (PLA).
Vegware for example do a full range.
But  there would need to be far more, large scale municipal composting schemes for this to be a properly effective answer but can check out this rather sweet cup to compost scheme here.

Biodegradable, Compostable Plastics

What is biodegradable? Biodegradable products break down through a naturally occurring microorganism into simple, stable compounds which can be absorbed into the ecosystem. More about biodegrading here

What is compostable? To be classed compostable, items must biodegrade within a certain time (around the rate at which paper biodegrades), and the resulting biomass must be free of toxins, able to sustain plant life and be used as an organic fertilizer or soil additive.

Composting Plastic At Home
FYI While most agree that some  plastics are indeed compostable, many say that they can only composted in large scale municipal schemes. I have used and composted a number of compostable plastic products 

More

Fooled again? Check out the lesser known sneaky plastics here