Well...used to be that if you were flying a published VOR procedure then the manuals very clearly stated "needles" took primacy and the fact you had LNAV was just a nice to have...nowadays with all the verbiage surrounding PBN I'm less inclined to be definitive without diving into the FCTM..
Then again I'll offer that my first response if asked that sort of question would be, "OK, lets be pragmatic, for example:
You are flying a VOR/DME with no RNP alerting requirement, and you get "Nav Unable RNP" .. the QRH may well will lead you to "verify position" - well to do that, lets look at the VOR/DME...
I