Definitely give it a shot if it's somewhere you think you'd enjoy. Again, it's great that you've got staff support, and such people are usually right if they tell you you're in with a strong chance.
I was exactly the same. I was so stone-cold convinced that I wouldn't get the postgrad offer I wanted that I was having pretty much daily collywobbles! In the end I was spoilt for choice, and I'm so glad that I got over my nerves and actually put the application in. You may have trouble with the acceptance letter, though, since I believe Cambridge's PG replies now come online through CAMSIS (correct me if I'm wrong?), so you might have to get people to verify a print-out, instead.
Well, you'll never know whether you're the best unless you apply. There's nothing you can do about it other than put in a strong application and cross your fingers that you like what you see - but if you let assumptions about the other candidates put you off then you'll never know.