Can the tune "Tenderly" be played in 3/4 time and 4/4 time? My sheet music shows 3/4 (waltz) time, but most jazz versions are played in 4/4 time.
Yes and the 3/4 one is the original and the 4/4 one is an arrangement. A lot of songs have different arrangements for different reasons which have many, many different effects on the resulting piece. You can make Mary Had a Little Lamb swing if you want to or make it more harmonically complex if you wanted to and that would be a different arrangement of that song.
In this specific case the jazz arrangement gives a lot more "emotion" which the change in time signature along with giving the melody more space helps accomplish this.
Listen for yourself and you'll hear elements from the original and how the jazz arrangement takes these elements and transforms some of them:
For more information of the possibilities of how you can rearrange a song and the effects, see this questions.
Yes. Another well-known example is Fly Me to the Moon, written in 3/4 but far more often played in 4/4, both as a slow or fast tune. Most tunes can be stretched or squashed to add or subtract a beat (or two!), but lots won't actually sound successful. I used to play Lullaby of Birdland in 5/4 which did work well. Often, just making the last note of each bar longer by one beat will do it, but to be really successful, the tune needs to be thought out referring to which notes sound better elongated. It's all good fun!
Edit: just heard a take on 'Take Five', played in reggae 4/4 timing, except the chord sequence for the verse stays on i rather than i, v in the original.
I say no and yes. The subject to your question is slightly different from the body of your. To be specific, a tune written in 3/4 cannot be correctly played in 4/4, but, as others have mentioned, you can arrange tunes successfully into whatever you want, to some degree. The reason I mentions this is that if you take your written sheet music in 3/4 and try to play it exactly as written but you attempt to count in 4/4 it will not work correctly as it was originally intended.
Yep - you get an odd in-and-out effect on the timing as for one 'round' the music/lyrics sound on the beat, then as the 4th extra beat is added things sound like they're offbeat, then back again.
I really like it.
Here's an example which most people will ahve heard: The First Noele by Steeleye Span (click the preview link)