Washer is plugged into bakelite adapter that has ground wire welded to copper cold water pipe.

I suppose that depends on whether the copper pipe is continuous all the way to the street (no plastic sections, filters, water softeners etc) and if the ground electrode conductor is attached properly. For years that was legal but sometime back around the Reagan administration they changed the code because of the prevalence of plastic parts in the water system.

"Reasonably?" Probably, but GFretwell has outlined the possible problem we have no way to know anything about...

Or more generally, if metal pipe goes all the way back to the main panel and is bonded to the grounding system there. And aside from the ground issue, what shape everything else associated with it is in.