Why can my Mac Pro early 2009 not maintain video output?

Hi. I bought this 4,1 2.66Ghz quad-core machine earlier this year, and everything seemed fine for a while. It came with a non-Apple GeForce 8800GT installed, and so it always booted with a black screen until the login interface came up. It came with El Capitan installed. I partitioned the drive and installed this Jan version of El Capitan in one partition, and Mountain Lion in the other, as this is where I have best access to my software, and this is my main boot. It came with 8GB of ram, which I upgraded to 16GB.

In the last couple of weeks the machine has refused to send a signal to my monitors, an NEC Multisync lcd2090uxi and its rebranded kin, a LaCie 320. The Apple chime sounds, and the machine whirrs happily away, booting up nicely, but no signal to the displays.

I found that I could prompt life to the displays by swapping round the DVI connectors, but as soon as I had to start-up, reboot, or wake the machine up from sleep it would be back to no signal again, and repeating the swap technique would then not help.

I am then in effect locked out of use of the machine.

I blamed the issue on the card and the vendor kindly sent me a replacement card...of the same model. The same problems have occurred, so my original card is not "faulty" per se.

The next way-in, I've found, was to swap the card to a different PCIe slot. This brought a signal back, so I wondered if slot 1 was failing. However, the cycle I've described before would then begin again. I found I could even replace the card back into slot 1 and get the same cycle of success and failure I've had before.

I've tried swapping the power cable to pcie aux b for both slots just in case aux a might've been failing, but to no avail. I've even tried reseating the processor tray, wildly-supposing there might be an issue there. All that achieved was a better seated processor tray.

I'll add that when I first had the machine I ran it to a LaCie Electron22Blue CRT with the LaCie 320 and everything was fine. I bought the NEC to replace the CRT behemoth, and the problems have started since then. The NEC was a cheap eBay find which I discovered has a power management issue, in that I could sent the 'puter to sleep but the NEC would stay warm with the screen faintly glowing, even tho' it told me that it was receiving no signal.

There is a part of me that wondered if this monitor might be the problem, but I don't see how as when it receives no signal it's telling me that, so surely it's not interacting with the card? The power management issue is a self-contained one, which just meant that I had to manually turn the display off, rather than depending on it going to sleep. Further-more, the issues occur even if the NEC is switched off.

Admittedly, I have not tried re-attaching the CRT yet. I am loathed to.

Looking online I see that there are issues with this model of card, but I can't see anything specific to my machine. It's either that or my PCIe assembly is the problem...or something else I haven't thought of.

BTW, I've tried zapped the PRAM... One chime? Which if I remember correctly is different to the old days which required three for zapping.

Any thoughts would be appreciated.