pc specs pls? cuz my first guess would be a dieing gfx.

using actual drivers?
Any problem in other games?

rename ur quake3 config in baseq3 and excessiveplus folder and look if problem persists.

Try to add other images maybe from other maps... also you might try to see if this doesnt happen in other games that use OpenGl!

This can either be a driver problem or .. some custom mod/pk3 in your quake 3.

Quake3 itself is not adding sprites in places like that..

Those aren't even in the triangle vertexes points..to see the q3 triangles and check this for yourself:

devmap poq3dm5
r_showtriangles "1"

And from what i see in the image its spawned randomly and it has a granulation size that doesn't depend of the space's depth from the scene => Either those are added directly on the projection matrix (like a custom hack which doesn't work properly when using the 2d methods for drawing on a above layer) or its a driver/hardware problem in which the projection matrix is somehow corrupted before its sent to the analog/digital converter and later to the monitor (from what i imagine that this is very very very very hard to happen but never say never)

So in short, my solutions are:
1. reinstall quake3 and retry just after that to see if the problem persists.
or
2. reinstall another driver for your graphic card.

It doesn't have anything to do with your monitor since you can screenshot it.. That means that the graphic card is having the same image before is on your monitor.

- Edit - From what i see in the second image, this is most sure a driver issue.. reinstall it and all should be fine.
- Edit 2 -
Another small possibility is to have a corrupted/ custom OpenGl Client Dll (opengl32.dll) file either in your windows system directory or in your quake3 installation directory (where is the quake3.exe )

Windows 95/98/Me      - C:\Windows\System 
Windows NT/2000       - C:\WINNT\System32 
Windows XP, Vista, 7  - C:\Windows\System32

My opengl32.dll file has this size: 713.728 bytes (697 kb)
and the File version is: 5.1.2600.5512 (you can see it in the file's properties)

If you have this file in your quake3 installation directory. Delete it! It shouldn't be there.
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epsislow