Defining the "right" chords to accompany a given melody is a slippery and subjective business. Most folk songs and other simple melodies work just fine with the basic three chords ~ often even with just two of 'em.

Now, within a passage where the chord doesn't change, the melody generally includes some notes other than the three which define the accompanying chord. Those who like to make things complicated sometimes name a new chord for every single such note, which may be technically correct but is certainly unnecessary.

For example, consider "Fishin' Blues" in the key of C, as typically fingerpicked on the guitar. I generally think of myself as "staying" on the C major chord for the first two full measures while picking out the melody on the two highest strings, playing each of those two individual strings open and also fretted at the first, third, and/or (optionally, for those who can stretch) fourth frets. To my mind, this entire sequence is played against the backdrop of ONE chord, the basic C major triad, but I have seen trasncriptions where a new chord name (presumably a correct name for a four-note chord) is provided for every single note of the melody; each of these chords, presumably, consists of C + E + G plus the current melody note.

On the other hand, of course, there are more complex tunes which pretty much demand a more complex harmonic accompaniment. While one can, I suppose, cobble together a basic three-chord arrangement for any tune, some tunes require the use of a fuller set of chords to convey the appropriate movement through the melody.