I would like to greet all of you. Your comments and suggestions are very valuable, because my "mother milk" was FORTRAN IV on IBM7040. Now, after retirement, I have Mathematica Home Edition and this is a different kettle of fish.
My experiments:
NIntegrate[Hold[If[EvenQ[Floor[t]], 0, 1]], {t, 0, A}] works
NIntegrate[If[EvenQ[Floor[t]], 0, 1], {t, 0, A}] does not work
NIntegrate[Hold[If[Mod[Floor[t], 2] == 0, 0, 1]], {t, 0, A}] works
NIntegrate[If[Mod[Floor[t], 2] == 0, 0, 1], {t, 0, A}] works too
It seems to me that problem is connected with function EvenQ as can be deduced from yours experiments too..