Stumble and backfire

I’ve been working on a problem with my Jeep for almost a month now and it’s driving me nuts. It’s a ’91 Cherokee 4.0L, AW4 tranny, 4WD. It idles rough, like its randomly dropping a cylinder. It appears to get worse as the engine gets hotter. It stumbles about once every two or three minutes. At part throttle it runs fine. During hard acceleration or under load (2800 rpm and above) it starts cutting out again badly and backfires. The spark plugs are getting fire, the air filter is clean and unobstructed and the injectors check good with an ohmmeter. So far I’ve replaced the cap, rotor, spark plugs and wires, MAP, TPS, O2 sensor, Cat, header, cleaned the inside of the valve cover and PCV front and rear (new ones installed with new vacuum lines) and idle air motor. The CPS was replaced 3 months ago (before all of this started). None of it has made any difference. Any suggestions?

Check the coil wire, with an ohm meter. Grab your rotor gently and lift, just to see how much play is in the shaft (up and down/left and right). There is a spacer between the gear and the shaft housing that wears out, can´t be good, with that much flutter in the distributor. After market coil is not going to work well, OEM is about all the volts a stock distributor cap can handle.
Hook a vacuum gauge up to the intake, needle should vibrate a bit, but not wag wildly. If the needle shakes a lot. Time for a compression and leak down test. I check my compression once a year anyway, write it down in the back of the owners manual, just to keep track of possible problems.
Never have figured out exactly what the cam position sensor does (good question for JNeary, but he hasn´t been around for awhile), but worth a test, if everything else has been checked.
Maybe have a look at the simple stuff, is 1 firing, on 1 at the distributor. Sure your firing order is right? Easy to get everything off one cylinder (usually doesn´t make much difference, but off by two cylinders will really mess with your timing). Is the lock (has it been cut) still on the bottom of your, distributor? If not it may be time to re-index.
About half the time, mine has started acting stupid, it´s turned out to be water/corrosion/oil in a connector someplace.
Double check the plastic tube for your MAP sensor, is it cracked, is it plugged into the right outlet at the throttle body?
Check the wire for your O2 sensor, hard to get to, hard to see, could be fairly burnt or cut. they sometimes flop on the header, exhaust manifold leaks sometimes blow hot gases on the wiring. Should/could show up as a trouble code in your computer. www.jedi.com/obiwan/jeep/dtc.html

A final thing you might check if 8Mud's suggestions don't pan out, is the wiring harness itself. I've had a very obscure harness problem that was somewhat similar to yours, which turned out to be a slightly resistive splice in the common (positive) wire to the injectors. It would start and run fine, then begin missing when it was hot, often waiting until it had been run, shut down and restarted. An ohmmeter reading between the source at the coil and the individual injector plugs revealed it. It's a long shot, and in the case above, the missing was pretty regular once it began, but if all else fails, it might be worth getting out the ohmmeter again.

An injection malfunction like this will not show on a vacuum gauge, whereas an ignition malfunction usually will, so a vac. test is a useful starting point.

By the way, the cam position sensor is what the computer uses to time the injector pulses. If that goes bad, no injection. I just went through this on a 93, and in that case, it wouldn't start unless I unplugged and replugged the sensor with the ignition on. Once started it ran fine, so I'm guessing that it would not cause your symptoms, but I wouldn't rule it out.

The wiring harness connector to the TPS has been known to go bad so you could try jiggling it to see if the engine's behaviour changes.

Dr. Dyno said:

The wiring harness connector to the TPS has been known to go bad so you could try jiggling it to see if the engine's behaviour changes.

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I had one with a resistance in the connectors for the TPS, cleaned them real good and things improved.
Had one that didn´t idle worth a darn often hard to start, CPS cable had flopped over onto the mainfiold and burnt/melted most of the way through.
A crack in the exhaust manifold, will cause it to pop at start-up. Has been said, it can also mess with your O2 sensor values, when the manifold is cracked, often a chug, chug, chug type miss at 2000-2500 RPM.
I had one with a serious miss, took me months to figure out, turned out to be a combination of a whole bunch of small things.

Put in a new CPS yesterday. Not any better, its still doing the same thing. Its not throwing any codes either. On a hunch I checked the fuel rail with a pressure gauge: 31 psi at idle, 39 psi when I disconnected the vacuum hose to the regulator. According to my manual that should mean that the fuel pump and regulator are good. I’ll check the distributor stuff in a couple of days when I have some time to play with it. Thanks for the suggestions guys, keep them coming.