How many here actually believe?

 How deep.

Yes, in this case "Not I" is correct, and "Not me" is incorrect.  To determine which to use, rephrase the sentence and see whether you would use "I" or "me".  He was answering that he did not believe he would be a GM, so his full sentence would have been "I do not believe . . . " and not "Me do not believe . . . ", so the appropriate short version is "Not I!"

Ronald Reagan always got those types of sentences wrong, sayng things like "They were very nice to my wife and I".  In that case you determine whether to say "I" or "me" based on how you would say it if you were the only one in the sentence.  They were very nice to me (not they were very nice to I), so the proper form is They were very nice to my wife and me.

In contrast, I had a good time (not me had a good time), so My wife and I had a good time is proper.

Point this type of thing out to people in public and you'll be a hated person!  

 Haha. If I keep practicing like I have been I will.

I think most people who posted here could become a FM -- but it would be so difficult that I'd readily bet against it :)  Same story for me.

I think if you truly embrace the game, the love for the game and the analytic skills necessary to stop, look, think and then make a move, that is how you embark towards the GM status that you mention. Play higher ranked players but play every day. Happy New Year!

Not many true believers here, it seems.

Chess skill is more esoteric than people give it credit for.  Just as a player new to the game won't understand why or how a tournament player can beat them 10 out of 10 games, an amateur can't understand what goes on when a GM is playing.  For an amateur to say "I believe I can be a GM one day" is silly in a practical sense.  It's good to have high goals, and some people can make it -- but there's no way to know so early in your career. 

It's like after learning how to add to claim one day you'll win a Nobel prize for a timeless math proof -- to be completely honest, the person has no idea what they're talking about when they say that.

Yes, it's about faith not about proof. 

Yeah, we dug up a few of those names in a past thread.  It turned out all the people we could find like this were IMs in their 20s (or similar impressive things) and then really late in life decided to go ahead and push that little bit extra for GM

But like you said, this thread is about believing!  I think it's a great goal for a player to have, and there are certainly people who can make it with a lot of hard work.

Well, there have been a few GMs who didn't even learn the moves until their 20s, so don't give up!

This may seem like a bold claim, but I think I can achieve BM by the end of the day.

Should we start a pool?

Just as anyone with normal physical dexterity can learn to play the piano, anyone of normal intelligence can learn chess and become a moderately strong player, if they put in the time and the work.  But like piano, becoming a titled player is like becoming a concert quality professional, it requires talent in addition to training.  "Basic" master's titles like NM in the USCF or CM in FIDE represent less than 2 percent of the players who regularly play in organized tournament, and the step from there to Grandmaster is essentially as large as the first step from novice to master was. 

To quote the great Sam Mussabini, "You can't put in what God's left out."

 To use the words 'Not I' is correct, though it is very formal.  'Not me' is also correct in this context, and is more informal.  In this context, 'Not me' is more suitable, though it's up to you.

I believe I can beat GMs. I know I can't.

They are not the same grammatically.

"Did you go out last night ? Not I !"

"Did that guy punch you ? Not me, he punched my friend !"

"Me" is correctly used to replace the object, not the subject.

I believe I could, but I believe I won't.  

I don't believe that.

I can think of a few possibilities.

1) Assassinate all existing master players. This should deflate the rating pool sufficiently for me to become a GM.

2) Get a tiny computer screen running Fritz 12 sewn into the palm of my hand. Keep this hand on my lap and consult it whilst playing.

3) Bribe someone important at FIDE.

4) Kidnap someone important at FIDE and demand the GM title as a ransom payment.