Blood pressure in the blood vessels
Answer: You are right that if you had one big pipe that was getting progressively narrower, the pressure in that pipe would increase. In the vascular system, you have one big pipe emptying into exponentially more small pipes. The total lumenal diameter of the small pipes is actually greater than the single big pipe.

Even though each individual capillary has a smaller lumenal diameter than each artery, the sum lumenal diameter of all of the capillaries is much greater. This is because there are way more capillaries than there are arteries.

More detail: The following image plots total cross-sectional area and pressure in the pipe against the level of the circulatory system. The sum cross-sectional area of the vessels is highest in the capillaries, simply because there are so many capillaries. You can see the pressure decreases accordingly.