Time Signatures

There is no such thing as a sixth note, therefore, a time signature of anything/6 cannot exist, and if you've seen a piece notated that way it is wrong. You might argue that a triplet can be considered a "3rd note", and I'm here to tell you that that is just something you've made up or someone else has made up that you've seen to comprehend the rhythm better. Notate it as a triplet, or better yet, make the time signature 3/8 and it should have relatively the same feel.

If you have a piece that's all in 4/4 with the exception of a single bar that has five quarter-note triplets, it's much easier to notate it as 5/6 than as 5/8 or 5/4 with a tempo change at an exact ratio (although I suppose that putting "[quarter note] = [quarter note triplet]" could be done). Writing a piece that's in 5/8 as being in 5/6 by just calling all of the eighth notes triplets is pretty stupid, but irrational time signatures do have their place in my opinion.