SQL left self-join with WHERE clause dependencies between the two copies of the table

Move the word_number + 1 requirement into the LEFT JOIN.

SELECT
  s1.word word1, s2.word word2
FROM
  sentence_words s1
LEFT JOIN
  sentence_words s2
    ON  s2.sentence_id = s1.sentence_id
    AND s2.word_number = s1.word_number + 1
WHERE
  s1.word_number = my_start_number

NECRO EDIT:

Although the above fixes the use of LEFT JOIN, I would suggest not using joins at all...

SELECT
  sentence_id,
  MAX(CASE WHEN pos = 0 THEN word END)   AS word1,
  MAX(CASE WHEN pos = 1 THEN word END)   AS word2
FROM
(
  SELECT
    sentence_id,
    word_number - MY_START_NUMBER   AS pos,
    word
  FROM
    sentence_words
)
  AS offset_sentence_words
WHERE
  pos IN (0, 1)
GROUP BY
  sentence_id