Is it possible to install two dual output PCI graphics cards for four total displays?

Yes. If you can find one then you should be able to install a second PCI graphics card to the run the extra monitors.

This is definitely possible with two PCI Express cards in SLI configuration, as I’ve seen it done. As you only have one PCI Express slot and several PCI slots, I can’t say for sure that it will work.
I’m very inclined to agree with @Lightlyseared here though, it’s worth trying and unlikely to hurt anything on a trial run. Of course the usual caveats apply: make sure you have enough system power available from PSU, watch your temperatures, etc.
Good luck!

It will be tricky, mostly since PCI graphics cards are a bitch to find these days; AGP replaced it around six years ago (my last P4 had AGP 8x, and it was built in 2004) only to fade over the last 2–3 years as PCI-E has dominated. Fortunately, that will be your biggest problem, though not your only one.

Odds are that any decent PCI card you find will be older technology, which means more heat and more power consumption. While you likely won’t have a pig like my old Radeon 9800 XT or @koanhead‘s nVidia space heater, I am not sure if your power supply is up to snuff. If your Optiplex is anything like my old Dimension 2400 then it probably isn’t unless you stick with low-end cards. But a PSU is cheap and easy to upgrade if you need to. Having fried a couple of PSUs in my day, I tend to err on the side of caution on that these days.

Cooling may be an issue, especially if you put two heat hog cards in close proximity, but like the PSU, it shouldn’t be a problem so long as you don’t get a power-pig card. Keep an eye on it.

To sum up, it is possible, and I second @koanhead‘s caveats, though I might be a little paranoid.

Thanks for the comments guys. I installed a second nVidia GeForce 6200 PCI video card. Since I already had the drivers installed, I didn’t need to install them again. At first I was getting a clone of monitor #1 to monitor #2, but clicking around in the nVidia system settings and Windows Display settings I was able to select monitor #3. After a system restart, I am good to go. And @jerv- you are right, these nearly extinct PCI cards were tricky to find and a bit pricey ($80 for a 256MB card). Comparable PCI-E cards were more than half the price. Also… these cards to no have a built in cooling fan so only time will tell how well they hold up, but my guess is since they are drawing little power, they will be fine for what I am doing.