The mystery of coconut tree
First it needs to be said that coconut trees are not true trees, but palms, whose trunks are made of stems which grow in a cross woven pattern. 

That being said - its true for any plant that no plant grows from the bottom up.  If you make a mark on any plant that mark will not rise much as the plant grows. 

Plants grow from buds at the end of their stems.  As the plant grows taller  its because the tissue adds from the ends of its branches/stems.  



In this illustration you can see the terminal bud, which is where the plant is growing up and also out into branches.  The palm is more like a bundle of grass stems, so the terminal bud is not obvious, but the leading edge of the grass will grow here. 

A tightly wound core of stems in the crown of the palm, called the Heart of Palm is where the palm grows (see "growth point" in picture below).  It is also a terminal bud, but it is not as exposed and is not as obvious. 


(figure from The Food and Agriculture Organization of the UN, fao.org)