I searched the Sapphire web site for your card -
Vapor-X R9 290 4G D5, with two DVI ports, one HDMI port, and one DisplayPort.
Like many video cards, yours only has the necessary hardware to support
two monitors at a time when connecting via DVI and HDMI connections. The card includes two DVI ports and one HDMI port for connection flexibility, but only two of those three can be used simultaneously.
DisplayPort connections don't require the same hardware support, so you can connect as many DisplayPort monitors as you have DisplayPort connections, in addition to the two monitors you can connect via DVI/HDMI.
Sapphire often sells monitors with
Flex in the title, which have extra chips to support more than two simultaneous non-DisplayPort monitors. Going forward, I will only buy cards with 2 or 3 DisplayPort connections + DVI/HDMI, since DisplayPort provides the highest resolutions and most flexible connectivity options.
Your card can technically handle 4 monitors if you get a DisplayPort monitor with both an upstream and a downstream DisplayPort connection and everything supports the DisplayPort v1.2 standard. You would then connect two monitors via DVI/HDMI, one monitor via DisplayPort, and a second DisplayPort monitor connected to the first monitor instead of the video card.
You should also be able to use a DisplayPort hub to connect 2 or 3 DisplayPort monitors to your single DisplayPort connection. If your video card does truly support up to 5 displays, this would be the only way it would be supported. Here's a sample product:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814998089
If you had more than one DisplayPort connection, you could buy an
active adapter to convert DisplayPort to another legacy connection that would enable an additional DVI or HDMI-based monitor. Active adapters require power, usually supplied via USB cable, and they can cost $25-50. A cheap passive DisplayPort --> DVI adapter won't do any good if you're already connecting two legacy monitors.
See the user R290 Series User Guide from the Sapphire web site, section 3.2.1 for full details.
http://asia.dl.sapphiretech.com/archive/gm/drivers/AMD_Radeon_R9_290_Series_635337573321470004.zip