Your question statement is a little bit incorrect or, better say, imprecise.
First of all it is not clear what you mean by the "body response" here? Pain is sensed by peripherial receptors (several types of receptors, mostly purinergic, are assumed to play role in pain preception) and then this sensation is propagated into CNS. So, the local "response to pain" (like swelling, increase of the blood perfusion) is not the response to the pain stimulus, but rather a methabolic response to the cell injures and release of many humoral factors like interleukins, "wound hormones" etc.
And this is true that out of many different pain inputs the one with highest intensity seems to "outshine" all other inputs and ultimately the pain perception is constrained to a single body part. However, if you consciously focus your perception on other parts of your body you may be able to percieve the pain originating from these parts even if this pain is lower.
This principle is known as a smaller particular case of a more general principle known as "the dominance principle of Ukhtomsky". Wikipedia has only the Russian language version about this principle (as well as about Ukhtomsky, a prominent Russian physiologist) and it seems to be not really known outside of the Russian scientific community (this is the only page I found about him in English).
The more general principle of Ukhtomsky postulates the following: one certain input may lead to the formation of the dominance center in the cortex. The domaninace center is characterized by the higher level of nerve cell excitation, whereas the rest of the cortex undergoes an inhibition so that different, even unrelated inputs converge here, helping this center to exist for longer time. The only way how this center can be ceased is to form another dominance center by a extremely high stimulus (for example a startle sound). Here the most insensive pain leads to the formation of such a center and all substantial pain stimuli just contribute to the persistence of this center: the pain is however percieved to originate from the place where the first stimulus, responsible for dominance center formation, originated from.