Spain and the small island state of Cyprus formally became the fourth and fifth countries to request aid from the euro zone's bailout funds Monday, extending a casualty list in a protracted regional debt crisis that is threatening to break up the 17-nation currency bloc.

Spain struck an agreement this month with the 16 other euro-area members to ask for as much as €100 billion ($125.7 billion) from European Union bailout facilities, specifically to recapitalize a number of regional banks that have been devastated by the explosion...