Like most grandparents in Italy, Isidoro and Antonietta Arcidiacone were thrilled to help out when their daughter, Grazia, and her husband started a family 2½ years ago. They got more than they bargained for.

The 67-year-old retired police officer and his wife have taken their daughter's family back into their one-bedroom apartment in Rome. Mr. Arcidiacone takes his two toddler grandsons to the playground and pediatrician. Ms. Arcidiacone makes homemade gnocchi and peels the skin off grapes so the boys don't choke. This summer,...