Hi, it is starting to sound like potentially one of your active adaptors is faulty or is somehow reverting to a passive adaptor mode.
If you disconnect one of the DVI > DVI monitors form the GPU, does the offending monitor attached via DP work ok?
I would expect that you get 5 displays working, 4x DP and 1x DVI from the GPU.
This could be due to falling back to a passive adaptor mode or the adaptor really is passive? and thus is using the output of the DVI connector - re-routed to the DP output.
You have stated that the monitor does work, and will work when connected to another output switching over cables.
This seems to indicate a fault with the adaptor in use for this display.
This behaviour is quite typical when using passive or faulty active adaptors.
A bit of background information as to why this happens.
VGA, DVI, HDMI and some other video signal formats use a TMDS (Transition-minimized differential signaling) chip to create a carrier signal for the video data to transmit.
The Displayport signal type does not require a TMDS carrier, as it uses its own protocol which delivers data in a packet method, much like a network switch rather than a constant stream of video signal like the signals listed above, however to do this it must use an active adaptor
Typically a GPU will have a single TMDS chip which has 2 or 4 channels, these channels are connected to the VGA / DVI / HDMI outputs of a GPU, although they can be re-routed through a Displayport output when using a passive adaptor to dvi/hdmi.
All DP>VGA adaptors are active.
Dual-link DVI will uses 2 of these TMDS channels, so on your card with the pair of DVI dual-link outputs all 4 TMDS channels are used.
Single-link DVI uses just 1 channel.
This of cause limits the GPU to driving 2 displays at any one time, with additional displays driven using the DP packet data protocol.
You may notice that the Sapphire Flex Edition cards are able to output to 3 displays without the need for any adaptors, this is because the 4 TMDS channels are split and shared between the 3 outputs, giving the user the option to display a 3rd display at the cost of max output resolution allowed.
When you disconnect the DVI output from the GPU, you also release 2x TMDS channels that may be re-routed to another output, if the DP>DVI adaptor is a passive adaptor, then it will use these 2x TMDS channels.
This could be why you are only able to get 5 of the displays working.
I would suggest you try pickup another adaptor and test with a new active adaptor to see if this helps.
I hope this background in the signalling system helps shed some light and understanding of the problem, which can be difficult to debug issues like your experiencing without understanding the greater picture.
Good luck and let us know how you get on!