Rain gutter backs up and overflows in heavy rain
Your gutter is overflowing because either the gutter itself is clogged up with debris or the down spouts are clogged up. 

Your larger sized gutter should be able to handle more rain than the smaller sized items installed on your neighbor's houses. So a smaller size would be even less able to cope with the problem than the size that you have. 

I suggest that rather than fitting new gutters it would be more cost effective to unclog the existing gutters and make sure that water can flow properly along the gutters to and then down the spouts. 

There is a slight possibility that some of your gutters may have been installed incorrectly such that they have a slope away from the down spouts. If this is the case then rain would gather and run toward the down slope end and then overflow because there is no where for it to go.