This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of Boolean functions in this model is very sensitive to the allowed error: There is an explicitly defined sequence of functions f n : {0, 1} n # {0, 1} such that f n has rectangle approximations with a constant number of rectangles and one-sided error 1/3+o(1) or two-sided error 1/4+2 -#(n) , but, on the other hand, f n requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant. Rectangle partitions and rectangle approximations with the same partition of the input variables for all rectangles have been thoroughly investigated in communication complexity theory. The complexity measures where each r...