I was under the impression that there is still oil deposits and thats is why teh war started, resources were becoming scarce, they had not dissapeared.

Well, assuming they run on fuel and not nuclear power, just before the war, oils and fossil fuels were bone dry in the world, so I would imagine that the planes would have been used by the wealthiest of the wealthy, the boomers obviously learned how to reproduce fuel, but we don't know how long they've been doing that for the last 50 years since they left Vault 34. I think airplanes arent something the NCR could repair, it might be to complex for them.

Do the Boomers mention making fuel? I always just assumed that Nellis as a military base had some fuel reserves left. Since McCarran was a civilian airport it presumably wouldn't have had access to the US's limited reserves.

If the boomers can use fuel, why can't the NCR? The NCR own some land in the wasteland.

If the boomers can use fuel, why can't the NCR? The NCR own some land in the wasteland.

Hot air Baloons - Probably exist in moderate and higher technology groups. Textiles and some gas burning technology required. Probably owned by richer eccentrics for pleasure, and used by organised governments (NCR, Vault city, etc) for scouting, cartorgraphy and aerospace research.

Lighter than air baloons and Zepplins - Helium aircraft (like blimps are today) are probably non existant due to the difficulty in getting helium. This leaves Hydrogen, like the Hindenburg used, which is plentyful and relatively easy to produce (Any water source, even if undrinkable + Electric Current = Oxygen and Hydrogen). Known to exist in the BOS (if Tactics is taken as Canon). Within technological grasp of the NCR, Shi, and Vault City, although the NCR is more likely than the others to build them it due to better resources. Can be used for transport, as well as scouting purposes. Probably out of reach of the regular citizen, although richer caravan masters may have an interest in this technology to get above the raiders.

Fixed Wing Gliders - Essentially an aeroplane without an engine. They can take off by various sources, such as another fixed wing aircraft towing it, a winch (essentually an engine with a long cable), an Auto-Tow, or a bungie launch from a hill. The Current standing endurance gliding record is over 56 hours, and was set in 1952 (new records stopped being existed at this point for safety reasons), showing that long term flights without an engine are possible given pre-divergence technology. This would require some degree of civil control (as an airfield of some type would be required, and would need to be maintained) and technological development (to build the aircraft structure and launch method). Gliders can be used for recognissance and transport (tanks and troops were transported by glider in WW2). Shi, NCR, Vault City, and the Brotherhood are all potential users of Gliding technology.

Fixed wing power (Aeroplanes). Some level of civil control is required to operate and maintain an airfield, and a good understanding of technology is required to build and maintain. AvGas (Fuel) is likely to be extremely rare, so any Fixed wing flying is probably limited unless alternative fuel sources are available. NCR and Vault City, with their access to pre war vaults (and the Vault library contained within) could possbily build a fixed wing aircraft. Brotherhood is a possibility, depending on the source of their library. Shi are less likely without these sources, but due to their organised scientific culture have the capability to rediscover the secrets on their own (if they cant find a source in an old San Fran Library somewhere). This is probably the minimum level of flight technology the enclave would show, however due to their lack of territorial control would have some trouble showing this power.

Helicopter / Vertibird. A Heliport is probably easier to control, operate and maintain than a full runway airfield, but helicopters would require a good technological understanding and capability. Enclave show this capability. NCR and Vault City with their access to Vault Library computers might be able to build some elementry helicopters. Shi and Brotherhood may have plans and capability to build vertibirds, depending on how you finished FO2. However, fuel is likely to be a problem.

Is it for the same reasons that Dr.lee didn't restore any of the planes on rivet city's flight deck?

Does the NCR have access to fuel? Yes, I am rather sure the Vertibirds are not Fusion powered

Could the NCR repair the planes? Odds are yeah given enough time. There are books everywhere odds are high a repair manual could be found if really needed. If not then given there amount of stuff to work with odds are they could figure it out with enough time.

Could the NCR learn to fly the planes? This is where I say no, and all the plans to use the planes fail.
1. They have enough to fix a few planes... sure, but there will be a ton of crashes, and you can only fix so much from scrapped items. I think they would run out of parts before they got a single man up in the air who could actually fly it.

Those planes the air-base are way, way, way more technical then the ones those that the Wright brothers built/flew.

You can't take a man who has never seen a computer before, and expect him to build, and run one out of 5 broken computers.