I'm not currently in the game industry, so these questions don't really apply to me right now. They might in the future, but who knows. Nevertheless I think about the following questions a lot, so I'll ask and see what kind of answers I get...

1. I've read plenty of times something along the lines of "Once you're in, it's easy to move to other positions within the industry"

Is that true? Why? If I'm a programmer for 3 years, what makes me more attractive as a producer than a fresh graduate? What do I know as a programmer, a different discipline, that anyone else wouldn't? Is it because I have 3 years in the industry? I've been working in cable advertising software for a year and a half, but I don't think that means I could be trained for a project manager position in my company any quicker than anyone outside the company could.

2. If number one is true, is there a limit to it? Is there a point where you just can't jump jobs any more? If you spend 15 years as a programmer and you are now a Technical Director, could you suddenly find yourself in a level designer position? (Why would you do that?) What about a Creative Director position? Why? What makes a Technical Director qualify for a Creative Director position?

The reason is that most of those positions are not entry level.

To be a good producer or game designer you need to know much about the industry. These are generally senior positions requiring years of experience.

Yes, it is possible for outsiders to step in and learn the ropes. Sometimes people outside the industry are brought in as associate producers or level designers. It is much more frequent that people who are already skilled at making games will transition to the jobs. Unlike many career fields, the jobs within game development often cross boundaries. Programmers and artists have a huge impact on design. Designers will often program scripts and make concept art. Programmers can work on artistic tools. Etc.

Having a programmer or artist transition into the more senior field of designer is more of a diagonal promotion; it is still creation, just creation of bigger things. It is not like the transition from a creation role to a people-management role.

Those who do transition into people-management also need to have a solid understanding of the game development process. They are often pulled from the ranks of other developers because they know the process well. They are usually people who have demonstrated skill at working through the development process and eliminating development barriers, at estimation, and prioritizing, and working with others. Again this is more of a senior level role and a diagonal promotion.

1.a. I've read plenty of times something along the lines of "Once you're in, it's easy to move to other positions within the industry"
Is that true?
1.b. Why?
1.c. If I'm a programmer for 3 years, what makes me more attractive as a producer than a fresh graduate? What do I know as a programmer, a different discipline, that anyone else wouldn't? Is it because I have 3 years in the industry?
1.d. I've been working in cable advertising software for a year and a half, but I don't think that means I could be trained for a project manager position in my company any quicker than anyone outside the company could.
2.a. If number one is true, is there a limit to it?
2.b. Is there a point where you just can't jump jobs any more?
2.c. If you spend 15 years as a programmer and you are now a Technical Director, could you suddenly find yourself in a level designer position?
2.d. (Why would you do that?)
2.e. What about a Creative Director position?
2.f. Why?
2.g. What makes a Technical Director qualify for a Creative Director position?

1.a. No. It's not "easy." It's certainly "very possible," though.
1.b. See frob's reply above.
1.c. Your proven natural leadership abilities make you more attractive as a producer. A raw grad thinks he knows everything -- he doesn't even know that he doesn't know squat. But you, 3-year vet, if you have shown leadership qualities and an understanding of the process and the priorities of the biz, will experience increasing pressure to take on escalating responsibilities.
1.d. Right. A year and a half doesn't amount to much.
2.a. Yes.
2.b. Yes. When you own the company, there's noplace higher to go. Except to sell that company and take over a bigger one.
2.c. You could. Not saying it's likely.
2.d. If you don't want to, don't.
2.e. Again, yes, possible.
2.f. Because the only things that are impossible are time travel to the past and the Star Trek holodeck. Everything else is possible.
2.g. Nothing inherent in the former position indicates the latter position. You've imagined a connection where there is none. If the tech director has been exhibiting spectacular creative talent on top of his technical wizardry, why would anybody want to deny him a chance to be brilliant?