Sometimes, I'm touching the blimp with no contesters around, and the point will not begin to get captured.
There are three KOTH variants that may explain this:
1. Straight KOTH- One point, everybody fights for it.
2. Crazy KOTH- There are several hills (A-E) that rotate between being active. While "A" is active you can only get points from A, and only take A. Once the timer reaches zero the hill moves to B, and then to C, D, E, and to A all over again. My guess is you were trying to take a non active hill in crazy king.
3. Resource Race- All hills (A-E) are active. Having more hills captured will accrue points at a faster a rate.
Also, no one calls it "the hill" but saying you get points from capturing points and the more points you capture the faster you get points sounds awfully confusing.