From Longman Dictionary of Contemporary Englishjam somebody’s/the switchboardjam somebody’s/the switchboardTCTif telephone calls jam the switchboard of an organization, so many people are phoning the organization that it cannot deal with them all Viewers jammed the switchboard with complaints. → jam
Examples from the Corpus
jam somebody’s/the switchboard• But look at the text, calls jammed our switchboard all day.• Residents jammed the switchboard of city authorities asking whether it was safe in the area. 