I'm an Admissions' Tutor for a programme that always goes into Clearing (we're a cash cow for the university...): I will ALWAYS talk to someone who's missed their offer, if they ring, even if it's been rejected through UCAS already, and I make this clear at Open/Applicant Days. If they can give me a persuasive argument (or their head teacher, in case of something that went wrong etc) I will go and make their case to the University's Registrar, and see if we can admit them anyway. It's a total ball ache, in that the Registrar is a total twat - but I have taken students this way more than once. Most of them who miss their offers, though, don't bother to ask - presumably because they think there's no chance. There's always a chance
I also missed my offer as an under-grad, and was told by my first choice that I couldn't have a place as a result; my insurance also rejected me. I found a place through Clearing, but in the meantime, wrote to the AT at my first choice and made a case for why I should still be allowed to go - and why I'd blown it: and, crucially, why I wouldn't blow it again given a second chance. I got my place
(Actually, if I could go back, I'd tell myself to hold firm against my parents and school, stick with my revised plans and do my own thing - I fucking hated the university, and loathed my three years there... but most people don't feel this way!)