Error in Jump & Run Collision

My guess is that it has something to do with this code:

; convert pixel positions To tile positions
	X1 = X / TILESIZE
	Y1 = Y / TILESIZE
	X2 = (X + Width) / TILESIZE
	Y2 = (Y + Height) / TILESIZE

Maybe you have to subtract 1/2 tile to compensate for rounding errors?

X1 = X / TILESIZE - .5

(Sorry for not knowing for sure...I'm very tired and can barely stay awake :)