What does it mean that a checkered map can be colored with two colors? Can you show an example? How would it be possible that the last map can be colored with just two colors?
"Some situation" really refers to the precise case of all the countries shaped as a square of same size except with one country having a longer length than the rest of the countries on the map, and if that country's length is longer by a multiple of even numbers (twice as long, 4 times as long, 6 times as long, etc) the 4 color solution fails.
So in this case, 4 colors is always "sufficient" as long as the length of one country is longer by a multiple of odd numbers (three times longer, five times longer, etc...), but 4 colors is not sufficient in case the length is longer by a multiple of even numbers (twice as long, 4 times as long...) and the solution may require 5 colors or 6 colors, or more.