Jets don’t actually have a V[sub]NE[/sub]. What they have instead is two numbers called V[sub]MO[/sub] meaning Airspeed - Max Operating, and a Mach limit, M[sub]MO[/sub] meaning Mach - Max Operating. At low altitudes the V[sub]MO[/sub] is more limiting whereas at high altitudes M[sub]MO[/sub] is more limiting. The crossover altitude varies by type but is typically around 27,000 ft. V[sub]MO[/sub] in a 767 is 340 knots indicated airspeed (IAS) whereas M[sub]MO[/sub] is M0.86.
At very low altitude, flight at V[sub]MO[/sub] = 340 knots IAS will correspond very closely to 340 knots actual speed through the air (TAS) or over the ground (GS), as ably explained by **Johnny **in post #4. If we take the OP’s 9/11 speed estimates at face value, 510 knots GS ~= 510 knots IAS is 1.5 times V[sub]MO[/sub] but is still only around M 0.75, well below M[sub]MO[/sub].
With all that introduction out of the way …
There are two main risks with flight above the limits: flutter above V[sub]MO[/sub] and local supersonic flow above M[sub]MO[/sub].
Flutter is chaotic motion of moving parts. Think like a sports team pennant attached to a car at 70mph whipping madly in the breeze. Usually it affects control surfaces or small fixed protuberances like antennas. Essentially the aero forces get stronger than the stiffness of the part and its actuators or supports can withstand & it begins to oscillate. They won’t survive but a few seconds once flutter sets in.
Local supersonic flow is more subtle. Once the overall aircraft is traveling above about 0.9 Mach there will be localized areas where the flow exceeds 1.0 Mach as it rushes around curves or sharp edges. The top of the wing inboard and the bottom of the horizontal tail are typical locations to first see supersonic flow as the speed increases.
If the area of supersonic flow gets big enough on the horizontal tail, the tail loses effectiveness & the nose drops uncontrollably. Which in turn means you pick up speed going downhill. This is obviously a positive feedback loop that can potentially run away and quickly lead to ludicrous speed going more or less straight down.
Compared to the two main risks above, a third comparatively minor risk factor is gear doors and fairings blowing off. These things are held closed or held in position by mechanical locks. Which are sized for the expected forces. If those forces are exceeded by enough, the lock fails and the door starts to open or the fairing starts to peel off. Once the relative wind gets under there, the part is ripped off almost instantly. Which may cause cascading damage if it hits something further aft such as rear-mounted engines or the tail. Big flat doors are more at risk from V[sub]MO[/sub] exceedances, whereas small highly curved fairings are more at risk from M[sub]MO[/sub] exceedances.
Overall, getting even a good margin above V[sub]MO[/sub] doesn’t scare us. Not that we treat the limit casually, but we recognize the airplane won’t turn into a pumpkin if some kind of upset recovery finds us temporarily over V[sub]MO[/sub]. The bigger danger in a V[sub]MO[/sub] exceedance is getting too antsy & exceeding a G-limit trying to get the nose back up to level too quickly. That way lies structural failure and catastrophic inflight breakup.
Conversely, getting even a smidgen above M[sub]MO[/sub] is a whole 'nother kettle of fish. Things get bad in a real non-linear fashion not too far over the limit. Thar be dragons. I’ve said this before, but IMO most attempts by lightplane or non-pilots to John Wayne a jet will quickly devolve into an M[sub]MO[/sub] exceedance & a runaway irrecoverable dive.