How can a tree fill a coconut with water? Water goes up in plants by cappilary action. So that should also explain water inside coconuts.
My doubt relates to the easy experiment of transfering water from a filled glass to an empty one using some chord or even a toilet paper pressed as a chord. 
Even if the paper is initially dry, water is gradually transferred until levels are equal at both glasses.
But it is not possible (at least I couldn't make happen) to transfer any drop of water to an empty glass whose bottom is above the water level of the filled glass.
So how is it possible for a coconut? Maybe its internal pressure is below atmospheric?
I know there are other similar questions, but the answers tends to follow the capillarity action (what is obvious to me, but seems incomplete), or postulates great negative pressures. (But if it is 1 atm outside, at most it should be zero inside, and a perfect vaccum inside plants is too strange).
But there is maybe some combined effect of capillarity and negative pressure that is not well discussed so far.
 A: 
Water goes up in plants by capillary action

This is a common misconception. Capillary action is believed to play a small role but fails to explain how water is carried to heights greater than 1 meter. Water in transported through the xylem vessels (water-carrying tubes) mainly through two processes-
1) Osmosis occurs due to the difference in osmotic pressure between the region inside the root cells and in the soil. Hence water flows into the roots to equalize the pressure difference which is actively maintained by the plant.
2) Transpiration from the stomata at the leaves leads to negative pressure being developed at the top of the water column. This is known as transpiration pull and it causes the water to rises upwards in the vessels.
This is the currently accepted model of xylem conduction in plants. Do not confuse osmotic pressure and air pressure as they are completely different. Most conduction in plants makes use of osmotic pressure caused by a combination of water potential $(\psi_w)$ and solute potential $(\psi_s)$.
