When I hear the phrase 'Gaussian Elimination,' I think of elementary row operations to solve some sort of linear system of equations. And when I think of when it 'does' or 'doesn't' work, two things come to mind. You either mean that the resulting system is solveable, or you mean that somehow the row operations will somehow ruin the system.
So I answer both. Gaussian elimination will never turn a solveable system into something unsolveable. It will also not change the solutions to any system. That's pretty magnificent, and not immediately obvious (at least, it wasn't to me when I first learned it - it was a bit like magic).
But some systems are underdetermined, unsolveable, or uniquely solveable. And Gaussian elimination doesn't change those, either. So there are systems where Gaussian elimination will not produce a unique answer. And in general, one must be a bit witty to classify an infinitude (which my spell-checker thinks is a real word - cool) of answers (usually involving kernels or images or parameterization or something along those lines).
So really, the only great thing about Gaussian Elimination is that it is highly algorithmic and doesn't really change anything about the system itself.