Precision Area Navigation (PRNAV) Approaches
In response to the original question, you can fly to B-RNAV/RNP5 quite satisfactorily with INS and DME/DME updating (Rho/Rho Navigation) In European airspace this is required for flight above FL100 generally - with some variation between contracting states. For RNAV approaches you should be using PRNAV/RNP1 or 1/3 and you won't get that accuracy without GPS. In European airspace, airways charts have been redrawn to WGS84 to ensure that waypoints coincide with the GPS coordinates.
In response to scrubba's comment, VOR and NDB are being phased out already (target is 2005!) but there is so far no intention to shut down the DME equipment that is co-located with most VORs and some Localisers. RNP5 will be with us for quite some time...
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Through difficulties to the cinema