Most cells are regulated by a system of checkpoints which decide whether or not they can proceed through the cell division cycle. At one checkpoint, metazoan eukaryotic cells (the kind in your body) must prove that they have sufficient space around them (are not too enclosed by neighbors) before it can proceed through the cycle.
Cancer cells are the one notable exception to this. They ignore 'neighbor constraints' and push on to proliferate their own kind almost continuously. Our bodies suffer accordingly.