The videographer says yes!
This photo is described by NASA as:
The Lunar Roving Vehicle (LRV) gets a speed workout by astronaut John W. Young in the "Grand Prix" run during the first Apollo 16 extravehicular activity (EVA) at the Descartes landing site. This view is a frame from motion picture film exposed by a 16mm Maurer camera held by astronaut Charles M. Duke Jr. While astronauts Young, commander, and Duke, lunar module pilot, descended in the Lunar Module (LM) "Orion" to explore the Descartes highlands region of the moon.
Image Number: S72-36970
Date: April 21, 1972
That date and Dr Sheldon's comment led me to the voice transcript, where page 726 starts:
Duke: ... That maximum acceleration?
Duke: Man you are really bouting[sic] it.
CAPCOM: Is he on the ground at all?
Duke: He's got about 2 wheels on the ground. It's a big rooster tail out of all 4 wheels, and as he turns he skids the back end, breaks lose[sic] just like on snow. Come on back, John. ... Man I'll tell you Indy's never seen a driver like this. Hey when he hits the craters it starts bouncing it's when he gets his rooster tail. He makes sharp turns. Hey that was a good stop, those wheels just locked.
Duke: The suspension system on that thing is fantastic.
CAPCOM: That sounds good, we sound like we probably got enough of the Grand Prix, we're willing to let you go on from here. Call that a Grand Prix.
Duke: Okay. Man that was all 4 wheels off the ground, there. Okay, max stop.
Young: Okay, I don't want to do that.
Duke: Okay, excuse me.
Young: They say that's a no-no. ... Okay. I have a lot of confidence in the stability of this contraption.
Duke: Go ahead I'm going to run back in.
Young: I knew you'd rather get out and walk.
Duke: That's right.
CAPCOM: After he saw the way you drove.
Young: Well, when Charlie's in here it's a lot less bouncy.
Even though the math claims no, for a smooth approach to a ramp with a sharp crest, ignoring suspension effects (see comments below).
The top speed it ever reached was 18 km/h, or 5 m/s,
under Gene Cernan's command.
Driving up a ramp, the angle to leave the ground is 2 arcsin (sg / v^2),
where s is the distance that the vehicle is airborne,
and where, on the moon, g = 1.625 m/s^2.
To get it barely airborne, set s to its wheelbase, 2.3 m.
So the ramp would have to be steeper than
2 * arcsin (2.3 * 1.625 / 5^2) = 0.3 radians = 17 degrees.
"On the basis of crew observations and photographic coverage"
it traversed slopes no steeper than 12 degrees.
"The maximum slope angle that could be negotiated by the LRV
has been estimated to be of the order of 18 to 23 degrees."
So unless leadfoot Gene deliberately floored it over the sharp crest of a hill steeper than he could expect to maintain control, all four wheels could not have simultaneously left the surface.