There's an example I recorded today for the N64 where it's possible to force it, but unlike the GC, games were never designed with widescreen in mind, so you can see all the graphical issues.
This is a problem because the emulators are rendering items outside the native display resolution. Games which will "unload" actors and entities off-screen will disappear when you use the fake widescreen rendering. This is nothing that can be fixed, excepting creating new hot-patches that will force the game to run less-efficiently by not unloading entities that are rendered off-screen; however, code such as this will likely be unique to each and every game and require much labor to perfect, let alone get working.
Various GameCube games also suffer from the larger widescreen rendering (such as Fire Emblem: Path of Radiance), as they, like the N64 games, were not necessarily all intended for widescreen gaming. (This isn't necessarily to call out the poster above, but is more to expand more as to why the problem exists, for those interested).
Now, I've actually bee curious: why can't one simply "squish" an image at the correct aspect ratio, so that when playing the games on a 16:9 display, the games will "unstretch" back into their relatively-correct aspect ratio? I've seen several games render the games with such a squished nature (Final Fantasy XII, Perfect Dark, Goldeneye 64) that didn't have too many problems with such a method. There will likely be 2D distortion to a much greater extent than if the games were purely 3D, but still, it seems to me that the games ought look better rendering with the method I'm thinking off, than simply stretching a correct 4:3 ration into a bloated 16:9 image (withstanding the normal native resolutions displayed by the system already).
Or something.