A couple days ago, I was playing "I'm a Believer" on my keyboard (figuring out the vocal melody both by ear and from memory), and I discovered something interesting, which I then verified by referencing the recording.
The song's in G major, but there's an F natural (the only accidental in the vocal melody) in the last line of every verse. It corresponds to "all my" in the line "Disappointment haunted all my dreams" and "I got" in the line "When I needed sunshine I got rain." This foreign tonality musically represents the "disappointment" and the rain in place of sunshine.