Do you know what species of palm it is ?. Some of the more robust palms can grow very tall, like those planted along the boulevards of southern california which are often referred to as "sky dusters" locally.
Other types can be much more squat in their nature and slower growing.
Others which are often referred to loosely as palms as palms may not be true palms at all, cordylines are a prime example, these can produce side branches (usually after flowering has taken place.
As previously stated, true palms grow continuously from the top and the dead leaf stems, form the expanding trunk, so to cut the top off would kill it for sure. Mature cordylines however can be reduced in height by sawing through the stem and should resprout below the cut or multiple sprouts from the base.
The hardiest of the fan palms is the Trachycarpus from china, which cannot be reduced in height. I have a 6ft "trachy" which I prune the lower fronds from each rear (not too close to the stem) and when I see the prices they sell for, makes me appreciate it all the more.