You could try to split them. I dug one up from the garden a few years ago, and it split into three, all on it's own.
If you tip them out and they don't split, they will always benefit from some fresh compost.

And a larger container...

Yes, they can be, though some varieties cope better than others, but in general, yes. If they're potbound, you will probably need a sharp breadknife to saw through the rootball.

I'm really glad you asked this. I grow lots of clematis and I had no idea they could be split at their roots. When I read your question I pulled a face because I thought Clematis didn't like root disturbance, and certainly I have generally found them hard to move successfully in the past. So until I read the answers above I would have said no...just move it to a bigger pot or plant it in the ground. So..please let us know how it goes..I'd love to know!

Thanks all for your comments and suggestions. Visited my local garden centre today and posed them the same question. The emphatic answer was no! So after considering buying new larger pots or 2 new plants, the
more cost effective option is to buy new plants. So that's what I'm going to do and I'll find a home some where else in the garden for the old plants. Thanks again, happy gardening!!

Excellent solution cats! Dont forget to water and feed them very generously! :)

I think that garden centre wanted to take some of your money.
They can be split and they can be moved. I've done it myself.
When I moved here I brought several from my previous garden quite successfully :)

Yep, I've done it several times too - never done a species clematis (such as montana or tangutica) but have with several different hybrids.