In reply to NorthernerDownSouth:
Probably - I booked about 2 months ago for North Lees, it's crazy (learnt lesson from last year)
What I did last year was spend about 2hrs driving round trying to find a spare place, and got lucky when someone was leaving Coopers in Edale early.
There's always the Robin Hood's Cave option for bivvying, although that will probably be full too!
I suppose you could always knock on a random farmhouse door and see if you can pitch in the corner of a field? Not sure if it's done these days, but when I was little my parents used to do this!