Technically, it requires a delay of 90 minutes or more. From the CoC --

"In the event of flight cancellation, diversion, delays of greater than 90 minutes, or delays that will cause a passenger to miss connections, Delta will (at passenger’s request) cancel the remaining ticket and refund the unused portion of the ticket and unused ancillary fees in the original form of payment in accordance with Rule 260 of these conditions of carriage. "

DL has no separate "trip-in-vain" policy that I'm aware of that would allow to cancel and refund a ticket simply because it is delayed by a few minutes. I would not expect that all agents are even familiar with the term. They do have the informal policy (not in the CoC) of allowing free changes for delays/schedule changes of more than 60 minutes.

"If a Delta schedule or routing change has delayed your departure or arrival by more than one hour, you may be eligible to select an alternate flight at no additional charge. Note that the below conditions may apply:
•Your origin, destination and travel date must remain the same
•Alternate flights must be available, and you can only modify once as subsequent changes may result in additional fees
•Voluntary changes to other flights not impacted by a Delta schedule change may result in additional fees"