Brake fluid level normal? See any leaks anywhere? Have you or anyone done any work on the brakes or on the car that might have affect a brake component?
Your symptoms point to low pressure in the system, when eventually the pump shuts off and you've got no power brakes.
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The brakes use a high pressure hydraulic pump. Normally, it pumps up the system in short spurts and shuts itself off until it's needed again..
In the event the system cannot maintain high pressure, the pump will run for about 20 minutes straight before it stops.. then it must rest for about 5 minutes. Then it will run again and try to raise system pressure. If it fails, it'll shut down again.
There are many potential reasons the pump cannot maintain high pressure in the system..
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Here was my experience after taking possession of a "free" LSC... the brake pump would overwork itself and stop working in city traffic (where you use the brakes a lot). Both lights came on. After waiting a while, the brakes were OK for a while.
The brake fluid resevoir has a internal filter. My filter got clogged (polluted brake fluid perhaps). Very little fluid was allowed into the pump. The pump would occasionally run dry and .. you know the rest.
A $6 junkyard brake fluid resevoir fixed my problem.. a problem Lincoln couldn't diagnose, btw.. I'm not bragging.. just giving you an idea of the oddball possibilities.
Any information you can provide might provide a vital clue.. don't be shy..