A Air Carrier crew flew the LGA Whitestone Climb Departure from Runway 31 even though the LGA VOR and DME which are required for navigation were out of service. ATC issued the clearance and the NOTAM.

Narrative

LGA VOR and DME were NOTAM'ed out of service; as noted on the ATIS which we did not notice. Takeoff clearance was the Whitestone Climb off of Runway 31. This requires a turn to 40 degrees at 2.5 miles off the LGA DME. No other DME is listed on the SID to make the turn. After noticing no DME readings from LGA; we made the turn based on prior experience in the good VFR weather ('wagged it'); with no problem; conflict; nor complaint from ATC. After this we noticed the NOTAM (and had already asked departure about it); and realized we (and many others!) had been given a departure SID that was impossible to follow. As SOP at LGA; I had read back the Whitestone Climb clearance to clearance delivery; and they had confirmed it. LGA clearance delivery failed to stop using the Whitestone Climb once the LGA DME was NOTAM'ed out of service; and came up with an alternate clearance instead; that in fact could be flown. If I had read the whole ATIS; I would have been alerted to question clearance delivery on the validity of our PDC clearance.