Of course an elf could stay behind and not go to the Grey Havens. Although, this seems unlikely since the Elves were fading from Middle-Earth.
Suicide is also an option for an elf, although it wouldn't get them much better. After an elf dies, his spirit (or fea) will go to the Halls of Mandos. From there, he will either be re-embodied in Valinor or he can sit around and think for eternity. If he killed himself, I don't think re-embodiment would be exactly what he wanted. Thinking doesn't sound much better.
But what Legalos said is true: An elf must be half-elven or have the grace of Eru (Luthien) to have the choice of mortality. In Luthien's case, the choice was hers in a way but the deed was Eru's. In Arwen's, the choice was hers alone, a "gift" that was the result of Elrond being half-elven.
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"And if you listen very hard/ The tune will come to you at last/ When all are one and one is all/ To be a rock and not to roll." --Led Zeppelin "Stairway to Heaven"
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