Bells Are Ringing (1960)

Ella Peterson is an operator for an answering service run by her cousin Sue. Lacking excitement in her personal life, Ella starts becoming involved in the lives of the service's clients, including a struggling playwright, Jeffrey Moss (Dean Martin). As Ella gets in over her head dealing with a bookie posing as a record producer she tries to hide her real identity. That becomes more difficult when she begins to fall in love with Jeffrey.