I made the mistake of not being specific enough on another thread just a few days ago. I'll try not to do it again.
The P1/PIC is always ultimately responsible for not banging into the gound. As a controller, the only time that I have to assign levels that are terrain safe is when I am vectoring an aircraft - at other times I assign clearances to a fix that are traffic safe and it's the pilot's responsibility to arrange the flight profile so as not to hit anything. It's a moot point whether I am required to issue terrain safe level and route clearances when I'm just watching an aircraft on radar.
But all of this essentially applies to IFR traffic - the original question was about a VFR flight. Under the normal straightforward, plain vanilla, VFR that the question appears to relate to, the pilot is entirely responsible for avoiding collisions with the ground or anything else. (Please don't ask about the other flavours of VFR!).
In the past I have known some units and individual controllers that have routinely issues clearances that cannot be complied with (at least, not without breaking some rule or other or applying a very liberal intepretation) but this seems far less common these days.