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It is not the 4k monitor. The newish Radeon 460 (W10) will display full screen.
My Radeon 290- (W7/64) displays the 3200x1800 with a black border.
Both cards have 4Gigs of ram.

Would love to have that extra screen space.
Maybe the 290 is just too old? (2014) (4K is full screen but that's too small)

Go into the AMD control panel and increase the screen scaling its a common problem..good luck

Yes to what vargis14 said. Especially if you connected the monitor using a HDMI cable. AMD used to underscale the output by default. I had lots of similar instances when setting up video in conference rooms at work.

Also I believe Windows 7, released in 2009, isn't as adept at the high resolution / DPI displays now more common. Windows 10 has better support for these new 4K monitors.

DisplayPort...
I looked into Radeon Settings and found "GPU Scaling" and "Scaling Mode"
Turning on Gpu Scaling and then changing Scaling Mode does affect 1440p,
"Full Panel" vs "Center".
But it has no affect on 3200x1800 at all. The screen stays "Center" with a border.

If I go into Control Panel > Display > Screen Resolution there is the 3200x1800 setting.
Continuing to "Advanced" > "Adapter" > "List All Modes" there is no 3200x1800 listed.

Oh, I tried a "custom" setting and it said that it was not compatible with the display.
(not the video card)... That is false since the 460 does display it. (on HDMI)

Thanks to all for the ideas!
Someday I will be able to buy a new videocard.
I bought the Radeon 460 $99(for W10)because I could not even get 1440p
with my older cards (the 460 does 3200x1800).

Probably Windows 7 quirks. If you swap the 290 to the Windows 10 system does it work?

EDIT: And yeah, using the scaling setting is a better choice than outputting a non-native resolution. Programs using standard GUI APIs will look the same, but those that are HiDPI aware will be able to take advantage and render 3840x2160 but with the sizing of 3200x1800. The only downside is that programs using custom GUI toolkits that don't support it will be 3840x2160 native sizing, and with the increasing popularity of HiDPI screens they're growing smaller in number.