If the two iterated integrals of a function f(x,y) that we calculate under Fubini's theorem are different, then that means the integrals of the positive and negative parts of the function are both infinite, and thus the double integral does not exist.
Say we had some function where the integral of only one of the parts of the function was infinite - e.g. integral of the positive part of the function was infinite whereas the integral of the negative part was finite. In this instance, the condition for Fubini's theorem:
$$\iint_{R} |f(x,y)|d A < \infty $$
would be violated. So Fubini's theorem would fail. However, the double integral would still exist (positive infinity). My question is: is the failure of Fubini's theorem independent of a double integral existing?