One possible interpretation of this question is competition between the cover crop understory and the primary crop (olive trees/grapevines.)
Another possible interpretation of this question is competition between a domesticated cover crop and a wild/native cover crop used in the understory.
In either case, appropriate choice of the cover crop species and subsequent management are key for avoiding competition with the primary crop.
The quality of wine grapes can be improved with competition, especially when cultivated in areas with higher rainfall and richer soils. Adequate stress to the grape plant increases the compounds which produce desirable wines. Cover crops can be used to achieve this with grapes.
Currently, most commercially-available cover crops for orchards/vineyards are perennials from temperate continental climates. For olive farmers, these types of cover crops are mismatched to the Mediterranean climates where olives are grown, and competition for water is an issue. Native annual cover crops can provide the benefits of cover cropping (erosion prevention, increasing soil organic matter, increasing soil microbial diversity, increasing water infiltration, improving water holding capacity, increasing soil fertility, hosting beneficial insects and wildlife) while completing their life cycle during the cool winter wet season and then senescing at the onset of summer, avoiding competition for water with the tree/vine crops during fruit fill and ripening period.
Management through well-timed mowing, spraying or light tilling of the cover crop can also be used to balance competition with the crop throughout the growing season.