It depends on the song, and the person covering it.
If it can be performed well by the artist who is covering it, then it
might be successful.
A song like "Witchita Lineman" would be difficult to perform as
effectively as Glen Campbell did. More versions haven't
turned out as well as ones that have.
(I'm speaking on a commercial basis).
If it's just an album filler or an indie group, then it wouldn't matter
all that much.

Some of the artists mentioned who have signature songs:
On one hand, some of those groups have material that has been over-done,
but the people covering those "signature songs" usually do a better
job at it. (Without mentioning who those artists are.)