What engine and year?
CPS(cam position sensor) is used for sequential fuel injection, if CPS isn't working computer will switch to Batch fire injection(if 4cyl then two injectors open at the same time, if V6 the 3 injectors open at the same time).
So really shouldn't cause a no start, but as adsm08 said it could, the 3.0l and 4.0l use the old distributor hole for the CPS, and this setup(the syncro shaft) also powers the oil pump, so.............if there was no CPS signal that could mean there is no oil pump, and computer may disable fuel and spark, but long shot IMO.
Especially if it runs then you have the no start when warm, failed CPS wouldn't fit that at all.
The 20psi fuel pressure would be a concern, pre-'97 should be 35psi, '98 and up should be 65psi
And that could certainly cause a no start.
ICM(ignition control module) and coil packs can have issues when warm.
Another long shot, TPS(throttle position sensor) can cause fuel to be cut off when starting, unplug it next time you have a no start, if it starts up replace TPS, but ONLY if it starts up.