Up until last year, Rebond Rumble, our team was just not a contender. We ranked in the 30s or higher almost every year and had yet to reach elimination rounds. I'm not really sure what changed, but suddenly we found success with #10 and #11 rankings last year in our regionals, and so far a #7 (top 8

) finish at Orlando this year.
Be certain, we are not heavily sponsored. Our school has really ramped up the engineering program and facility we use, but in terms of mentors and outside help, we don't really have a lot of that.
I think two things changed us. Simplicity and a good core group of students.
First simplicity. I feel like our whole robot in 2010 was riddled with motor driven acme-screws and slow, complex parts. Keep it simple! In 2011 we used a simple wheeled basketball shooter and sliding arms for the bridge. Nothing too complex, but still a good design challenge. This year, we've gone with the over-used two-wheeled frisbee shooter, and also a simple chain/sprocket 30-point climb. All of this works! No crazy contraptions or time-consuming apparatuses. We may not be getting a design award, but we're getting the job done and it's working.
Also, we've had a good core group of students who work hard. It hasn't always been a lot of kids, but when you have a handful that works hard and has dedication, things will work out.
I think any team can get into the top tier of FIRST. It may take a few seasons, but slowly any team can improve and start winning regionals and awards, etc.
Oh, and mecanum wheels are nice.