Depends on the range officer, and your definition of rapid fire.
The Lincoln range that Gewehr98 mentioned used to have a problem with allowing any "rapid fire" with any rifle above a .22 caliber.
I doubt that, seriously. I don't know how many times I had my Bulgarian SLR-95 up there, chewing up 50 and 100 yard targets, as well as my steel swinging owl target at 100 yards. I never caught any flak from the range officers, and I was letting them go pretty quickly. We never had a "X number of shots per second" rule. We did have an "all shots must impact the berm" rule, however, and for a damned good reason.
Rate of fire isn't the big deal at Lincoln. Accuracy of fire is. During your "rapid fire", were each and every one of your shots hitting the targets stapled to the cardboard backers on those wooden frames? Or were the wooden frames getting torn up, with bullets hitting everywhere but your target?
Lincoln's range is built in front of a quarry, which sits behind the 300 meter berm. Residential sprawl is also getting darned close to that range, never mind the pasture immediately to the right side of the range, with cattle grazing there quite often.
As a former range officer there, I'm well aware of what the rules are. I gave a person one warning if they were trying to spray bullets hither and yon, and I saw rounds bouncing off the ground and over that last 300 meter berm towards the quarry or neighborhood back there. There was no second warning.
Bump-firing, and firing from the hip were also not allowed. Shooting at the cows next door was not allowed. Shooting while being ????-faced drunk was not allowed. Shooting at the flock of wild turkeys which decided to walk across the range was not allowed. Handling firearms while people were downrange inspecting their targets was not allowed. And yet, people still attempted to do so (the club had to pay the farmer for the dead cow) and bitched about what kind of ???????s the club and range officers were when they were told they couldn't. Remember, you aren't the one paying the liability insurance on a firing range.
Rapid, controlled aimed fire was just fine with myself and the other range officers. It was always a safety thing that caused folks to get the range officer's attention. Sometime when you're out at Dillman Range, go ahead and look UP from the benches. See those pinhole spots of light in the roof? Guess how they got there, I'll give you a hint, it wasn't controlled, aimed fire...
I think that Lincoln place must be a club for hunter/benchrest snobs.
And that's total BS. Have you actually been there to shoot? Lincoln's members and public patrons are some of the nicest, most widely varied cross section of gun owners there are. Lincoln is also a very SAFE range, and in the name of safety, of course the club officers and range team aren't going to allow indiscriminate bullet hosing - there's no better way for a rifle range to get shut down by the local community. But that's a far cry from just plain banning rapid fire. Assuming folks know the difference, that it.