Thunderbird still exists today. It's a global business school -- one of the best, actually. The previous poster says it was filmed in Mesa and Deer Valley, but many (if not all) the real flying footage is at Thunderbird. Thunderbird Field was in Phoenix at the time, now it's in Glendale, Arizona.
You can clearly see the Control Tower (still on campus, and currently in a renovation campaign), and the aerial view in the beginning is definitely Thunderbird campus, which was designed to look like a bird from the air.
Thunderbird has an interesting history as an air base. When the war ended, the General in charge of the base bought it from the Government (for $1, I believe) and turned it into a business trade school that specialized in international management. Lots of students were former airmen and GIs on the newly-created GI Bill (1944), and Thunderbird went on to become a renowned school of international business.
Thunderbird the school is a wonderful place, and though this movie isn't an award-winning blockbuster, it's nice to see the great old footage of Thunderbird Field, the barracks, the Tower, etc. in a nice quality video. Thanks to Hulu for it!