NetRadiant as well as Xonotic handle textures up to 8192^2 pixels, greater ones aren´t displayed anymore or even makes the game to crash, though Q3Map2 still compiles the map.
The problem is that the bigger the resolution the bigger will be the filesize of the image. This affects the loading and compile time plus downloading your files will take ages. I made a test image (greyscale, so coloured images will be even bigger!) and wrote down the file size of each:
While huuuuuuuge textures are possible, don't go there unless you have a good reason too. As mentioned they increase file size and load time, but more importantly they eat up GPU memory needed for other things. Most things are fine at 1024^2, you could go as high as 4096 for specific things, but at that's mainly necessary when skinning (eg using models). For brushwork using a lower texture scale and multiple textures is the way to achieve high detail. Also note that if you're looking for max texture sexyness, begin with making sure you turn off all compression options:
Set all those to 0 and do a vid_restart. Be warned though, unless you have a gpu with tons of onboard memory, even the default maps and assets will likely mean constant dipping into system memory which in turn means lower performance.