Well, it takes me a gradual increase of effort over time to get up near my maximum heart rate. For me, the heart rate sort of lags the effort a bit.
Now it also seems to me that maximum heart rate is exercise specific. In other words, the same person will get different maximum heart rates doing different aerobic activities like running, cycling, rowing etc. Finally, I do not think that maximum heart rate is an indicator of fitness. I did not fool with this when I was out of shape. But my maximum heart rate does not seem to change with fitness level. (I have not done a study on this though). I believe that one's resting heart rate does reflect aerobic fitness though.
The body has a maximum HR that will decrease with age, is completely genetic, and does not increase with fitness. However, the maximum HR you will achieve will differ depending on activity, and may increase with fitness. Minimum HR is largely genetic and not really related to fitness. Seems to be related more to the amount of muscle the heart has to feed with blood. I have a much higher resting HR (high 30/low 40) than my lowest (32). I am far more fit now, and I weight about 20 lbs. more than I did then, mostly muscle...