Apparently not. I have questioned this old wives tale for some time now, seeing them occasionally off and on throughout the winter. Robins do not necessarily migrate south in the winter, but rather move to another area where food is more plentiful. Their diets change in winter. Omnivorous in summer, feeding on earthworms and bugs from the lawn and garden, they become vegetarians in winter, feeding on berries and fermenting fruit in bushes and trees.They are still around, just not as visible to us.
