I have a 2004 Neon that only starts sometimes. I have replaced the starter and injectors. When it does not start, it turns over, but will not start. If I use some starter fluid, it starts right up. I can run it for a while, shut it off and come back a couple hours later and it will start. Try again the next morning and it will not start.

 

Sounds like the check valve in the fuel pump is leaking back, the fuel pump then has to start from zero when starting after being off for a while.

Here's what the manual says.

DESCRIPTION
A combination fuel filter and fuel pressure regulator is used on all gas powered engines. It is located on the top of the fuel pump module. It contains a diaphragm, calibrated springs and a
fuel return valve. The internal fuel filter is also part of the assembly.
OPERATION
Fuel Pressure Regulator Operation: The pressure regulator is a mechanical device that is calibrated to maintain fuel system operating pressure of approximately 400 kPa ±34 kPa (58 psi ±5 psi.) at the fuel injectors. Fuel is supplied to the filter/regulator by the electric
fuel pump through an opening tube at the bottom of filter/regulator. The fuel pump module contains a check valve to maintain some fuel pressure when the engine is not operating. This will help to start the engine. If fuel pressure at the pressure regulator exceeds
approximately 58 psi, an internal diaphragm closes and excess fuel pressure is routed back into the tank through the pressure regulator.

 

You can purchase an inexpensive fuel pressure gauge at a local parts store. Screw it on to the fitting on the fuel pressure rail and let us know what the pressure is.