can I combine electrical cords and their dimmeers?
Short answer is no, not a good idea to do this.

First, splicing cords is a bad idea as they become fire hazards.  A splice drastically weakens the cord possibly allowing the conductors to separate which then will cause sparking.  The other, is the dimmer.  The manufacture of the lamp most likely used the cheapest dimmer possible to run the lamp.  Putting three lamps on one dimmer will most likely overload it.

I recommend looking for a new lamp that provides the lighting you need in a single package.