Why are some 32-bit PC games not supported on 64-bit operating systems?

32 bit games can run 64 bit Windows, typically without any issues. Older 16 bit games designed for Windows XP or earlier will not run on Windows Vista/7 64 bit.

Many times, software and games will claim they do not run on a particular operating system (Windows Vista 64 bit in this case) simply because they don't want to allocate resources to test on that particular system or support it. According to this forum post, Medal of Honor: Airborne runs just fine on Vista 64 bit.

For some games (not all unfortunately) that worked well in previous Windows versions (e.g. Windows XP), in Vista or Windows 7 you have the possibility to start the game in compatibility mode.

Just right-click on the executable, select Properties and in the Compatibilty tab choose Windows XP and try. Same for installers.

You can also try to run the software in Windows 7's XP-mode, which simulates a complete and independent Windows XP within your machine.