It wouldn't be required for any case where there is a way to define the Missed Approach Point (MAP) without DME. Ways to do that include:
- Timing (as noted in the question)
- If the VOR station is the MAP
- If another NAVAID can be used to identify the MAP (which would require a receiver for that NAVAID, in addition to the receiver used for course guidance). Examples of this could be cross-tuning a radial from another VOR station (so a second VOR receiver required) or passing an NDB (so an ADF receiver required).
- If the approach procedure specifies that radar can be used to identify the MAP. This would entail having the controller tell you over the radio that you've reached the MAP, as observed on the controller's radar scope.