1991 grand marquis was cutting off more and more and had some difficulty starting it until it finally wouldn't stay running and than wouldn't start at all. What could cause this?

It could be an electronic component failure, fuel delivery failure or engine internal problems. Fuel pumps often fail a little at a time. Pressure can start to fall as the pump wears and the engine will run worse as time goes on. A fuel filter can act the same way, pressure would start to fall as the filter gets clogged, and sediment would settle down after the engine dies, allowing the fuel to get throught the filter briefly next time you try to start it. To check the fuel system, see if you can start the engine using a little starting fluid. If it runs briefly while you spray starting fluid into the engine air intake, you probably have a fuel delivery problem. Check the fuel pressure, most vehicles have a fuel pressure test port so you can verify fuel pressure. If you have no fuel pressure, try replacing the filter first, especially if you haven't replaced it for a while. The pressure regulator can also be a problem and should be tested. Electronic components often fail when they've been operating for a while. After they cool they can work again, but the point of failure is a weakness and the electronic component can completely fail and not operate again. The electronic components in your vehicle might include the ignition system; coil, breakerless ignition components and other related components. Check for spark by pulling one sparkplug wire, connect it to a spare sparkplug the set the plug on a solid piece of metal and see if you get spark when you try to start the engine. If you think the fuel and ignition are functioning properly, run a compression test to see if the engine is mechanically ok.