In short, grading every 3 months as a lower Gup is fairly normal. You will begin to be able to tell if you're in a "belt factory" if you look to the higher Gups and Black Belts and wonder how on Earth they have X Belt.
I'm an instructor for an ITF-based Taekwon-do school and the approach that our umbrella organisation follows is to have gradings four times a year (so every three months). For the Gups, especially the lower Gups, your gradings will come thick and fast over the first two years or so. With adults, we generally double promote to 8th Gup at the first grading if their focus and basic skills are there. Once you start getting to Blue and Red Belts (4th Gup and up) your gradings will very likely slow to every 6 months or more.
But as @slugster mentioned, your pace is based on your wishes, your abilities and your instructors confidence in your collection of the skills (and training 2-3 times a week consistently will definitely help). Once you get to Black Belt, the gradings are much further apart. Our organisation prescribes 3 years between 1st and 2nd Dan, for example.
As an instructor, we look for students who are confident at their current level, but are by no means perfect. You won't even come close to "perfecting" the 9th Gup pattern until you are a Red Belt at least. I'm two weeks out from my 2nd Dan grading and I wouldn't claim anything close to "perfection" on Chon-Ji!
One of the greatest things about Martial Arts IMHO is that it's a team effort to advance your individual skills. Unlike a team sport, if you "suck" you're only holding yourself back and not the team. But as it's based on you as an individual, so is your progress. So even if you cannot kick at head height or jump you can still improve your skill-set and still advance in the art.
The most important thing about any Martial Art is to find an instructor that you trust and helps you achieve YOUR goals rather than their own (such as pushing you into competition if your focus is self defense). Find a good instructor and maintain your own focus and you will go far.
Best of luck!