No, unfortunately. Modern carpets are made from a variety of substances, some of which are natural and some of which are not. That is why carpet retailers have to send their off-cuts to land-fill, not recycling.
Eventually the carpets I am using will become so worn that they will be unsuitable as a mulch. That won't be too much of a problem for me - as a private individual I can take them to the council dump and won't be charged for their disposal. At least I'm getting some use out of them before they go to the dump and I'm also helping some very nice tradesmen save a bit of money. I'm doing my bit for the economy (well, for carpet merchants and the NHS, if not for green grocers).