Yes I think the title "Break, Break, Break" is appropriate. The poem opens with an image of waves breaking on the shore. The shore is cold and gray, and the poet struggles with how to express the thoughts rising up inside. The opening stanza suggests that the subject of the poem is going to be bleak.
The word "break" has more than one meaning, and so acts as a double entendre—with a double meaning—throughout the poem. Waves break and hit the shore, but hearts break as well from grief or sad events.
Near the end of the poem, in the penultimate stanza, we learn that the "break" of the title refers not just to the waves but to a feeling of heartbreak. The narrator states:
O for the touch of a vanish'd hand,
And the sound of a voice that is still!
Like the breaking of the waves on the shore, the speaker's sense of heartbreak repeats. Therefore, the repeating of the word "break" in the title is appropriate both to the waves and the speaker's sense of never-ending sorrow.