I know it is much desired by some, but the Secondborn have the Gift of Men, which is a blessing in it's own right.

[He pauses.]

There are however, a few occasions in which those of Elven blood have become mortal. In particular Lúthien Tinúviel. Then there are the Half-Elven, who were able to choose which path they wished to walk. For example My Lord Elrond chose to live as the Eldar do, while his brother Elros walked the path of Men.