According to Wikipedia, the Lunar Landing Training Vehicle had a "service ceiling" of 6,000 ft (1,800 m). I assume this is the maximum altitude that it can perform at.
What was the limiting factor?
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Blimey, that number surprised me since it only ever flew up to about 500 ft! Service ceiling is the maximum altitude (air pressure) at which an aircraft is designed to perform with expected stability that's constrained within its flight envelope:
In aerodynamics, the flight envelope, service envelope, or performance envelope of an aircraft refers to the capabilities of a design in terms of airspeed and load factor or altitude. The term is somewhat loosely applied, and can also refer to other measurements such as maneuverability. When a plane is pushed, for instance by diving it at high speeds, it is said to be flown "outside the envelope", something considered rather dangerous.
So this is about rating the vehicle according to its design limitations, altitude being one of them, defined more specifically by density altitude. Still, 6,000 ft? Whoa!
Why is 6,000 ft (1.8 km) an altitude they would have never flown the Lunar Landing Training Vehicle (also dubbed The Flying Bedstead) becomes more apparent when watching this Discovery Channel's video of the Neil Armstrong's LLTV accident when one of the thrusters failed and he had to eject before LLTV crashed and exploded on impact:
Adding more unpredictable weather at higher altitudes to the already rather unstable craft that had to be 100% operational to sustain flight would only mean such accidents would have happened more frequently, unnecessary risking lives of Apollo astronauts.
Still, one thing to keep in mind is that these 6,000 ft are altitude above mean sea-level. While the Lunar Landing Research Facility in Hampton, VA has an elevation of about 10 ft (3 m), they were tested at Edwards Air Force Base that's at an altitude of 2,303 ft (702 m). That constrained service ceiling there to a maximum downrange of 3,697 ft (1,127 m), which is a whole lot more reasonable given maximum flight altitude above ground they ever achieved.