I am not sure it is safe to say "it is not possible" to add or remove more than three electrons from an atom. For example you can have the compound
Tin(IV) phosphate. In this form, the tin has a +4 ionic charge and therefore has loss 4 electrons.
I think it would be better to say that representative elements tend to lose electrons that move down to the closest noble gas configuration (metals) or gain electrons to move up to the next closest noble gas (nonmetals). For most representative elements, due to their location in the periodic table, do not need to gain or lose more than three electrons to accomplish this.
One could also argue that the energy required to remove (or add) each subsequent electron is more than the previous one and eventually the energy required to remove the electrons are not compensated for by the energy released when forming the new ionic compound and thus for energy reasons we do not see the loss, or gain, of more than three electrons for well behaved representative elements.
Other than the elements closest to He, by atomic number, elements all want to have a full octet for chemical stability. Elements, like C with 4 valence electrons are too large to be like He so C needs to fulfill the octet.
If carbon were to donate all four of its valence electrons, it would be extremely unstable and would take an enormous amount of energy to remove the electrons in the 2s orbital.
Now, to think about carbon adding four valence electrons to satisfy the octet rule, you must look at the electronegativity of carbon. This value is 2.55, and when compared to other nonmetals ths value is low. With a low value, the element does not have the ability to take on extra electrons, but instead, take on partial ownership of the electrons in the form of a covalent bond.
When carbon forms a compound with 4 atoms of the same element, like CH4 are CCl4, both types of elements are nonmetals meaning that the bond is a convalent bond, by definition.