My .Net Winforms application creates three OpenGL rendering contexts in my main window, and then allows the user to popup other windows where each window has two more render
The best bet is that there is no real answer to this question. It probably depends on some internal limitation of the driver, hardware of even the OS. Something you might want to try to check is the number of available texture units using glGet(GL_MAX_TEXTURE_UNITS)
but that may or may not be indicative.
A common solution to avoid this is to create multiple viewports within a single context rather than multiple contexts in a single window. It shouldn't be too hard to unite the two contextes that share a window to a single context with two viewports and some kind of UI widget to serve as the splitter. Multiple windows are a different story and you may want to consider completely re-thinking your UI design if there is an actual need for 26 separate OpenGL windows.
It's hard for me right now to think of a real UI use case that would actually require 26 different OpenGL windows operating simultaneously. maybe another option is to create a pool of say 5-10 contexts and reuse them only in the windows (tabs?) that are currently visible to the user. I didn't try it but it should be possible to create a context inside a plain window that contain nothing else and then move that window from parent window to parent window to whichever top-level window it is needed in.
EDIT -
Well, actually it's not that hard to think of one. The latest Chrome (9.x.x), supporting WebGL may want to open many tabs each with a WebGL context... I wonder if they handle this in any way. Just tried it and ran out of memory after 13 tabs... That would actually be a good check for you as well to see if its something you're doing wrong or if chrome and firefox (4.0.x-beta) have the same problem