On my 1999 Linconn Navigator the Back end hs calapsed and the truck bounces up and down when driving,and the air pump does not engage

check your cv joints

YOU MAY HAVE A CALAPSED BIO FILTER WHICH MAY BE BLOCKING THE DUCTING NOT LETTING COOL AIR BY, YOU MAY HAVE TO TAKE DOWN THE A/C BLOWER ASSEMBLY TO FIND THE BLOCKAGE AND REPLACE THE FILTER.

You must have a leak or its not pumping air like it should. This happened to my dads Lincoln towncar back in the day.

Most likely the seals in your air suspension..Not a cheap fix. unless you can do it yourself.

THE NO 12 REASON FOR THIS IS THE AIRBAGS FOR THE SUSPENSION ARE LEAKING, U CAN REPLACE THEM WITH A NON AIRBAG KIT. GO TO THE WEB LINK FOR DETAILS.
http://www.suncoreindustries.com/english-lincoln-navigator-03-06-air-suspension.html

This is an expensive repair sometimes and you may want to replace this system with inflatable air bags that fit inside the rear coil springs and then are inflated. I think they cost about $80.00 or $120.00 depending on the car.

Remove battery, remove windshield washer bottle, remove air compressor and disconnect both air lines from the compressor dryer. Make sure compressor electrical wiring is plugged in, install battery, turn suspension switch to ON, turn ignition switch to the ON position, close all door and the compressor should come on and completely lower the whole truck. Once the truck is completely down, turn the ignition OFF, remove battery and connect both air lines. Install battery ensure electrical wiring is plugged in, turn ignition switch to ON, the compressor should come on and raise the vehicle to the correct ride height.

Does the pump run? Check the pump fuse or air ride fuse.

SOunds like a broken shock, or bad strut. Air bags or not, struts and shocks are what control bounce.

sounds like a bad air comp i get them used from the junk yard all the time