This is something I've been thinking about and trying to clarify in my understand of music. If you disagree or know more, please correct me. Outside of drums and percussion (non-tonal instruments), instruments tend to play three roles in music.
I ordered them like that to imply the generally high to low order of them, but it is not uncommon to find chords above the melody (like ska or reggae with a male singer) or chords and melody existing in mostly the same space. I will start with elaborating on bass, though.
Bass - this role is generally taken by an instrument capable of playing the lowest pitches such as tubas, doublebasses, bass guitars, and full-size keyboard instruments. The bass has the power to dictate what the chords are. For example if a higher instrument such as a guitar hits an Am the bass can confirm that by starting with an A or the bass can invert that chord by starting on a C or E instead (standard inversions, not the only possible ones).
Aside from this as some researchers discovered an obvious truth, bass and only bass is capable of creating a feeling of movement if desired. This might bring up the thought of the classic walking bassline or the triplet-based shuffle feel. As far as I know bass adheres to the same rules as melody with these added powers or responsibilities.
Bass is usually one note at a time because when you descend past C two octaves lower than middle C (string 3, fret 3 on a 4 string bass), these notes don't blend into chords and intervals very well. You can still get a decent sound out of the 5 interval, but that's about it. That generally cuts off at the A or A flat.
Chords - This aspect of music tends to be steady and stable. Chords usually stay the same for half a measure or more and generally don't change faster than a quarter note rate to avoid that feeling of too much shifting and changing. They are often ringing or repeatedly struck providing a point of stability in contrast to basslines and melodies that keep moving all over the place.
Aside from this they make the melodic foundation readily apparent. If you're in pure E minor and the bass is hitting an E with only a melody going, that alone can imply a foundation of an E minor chord (although, with just that I believe it could be used as a chord inversion depending on what melodic notes are doing among other factors). Although, an E bass note by itself with no determined key leaves the possibilities open for just about anything centered around E.
Together the bass and chords provide the foundation for melodies. If you were to hit the melody C, B, A with C and B as eighth notes and A ringing out, you could create several different feelings by doing it with a different bass note. A would be an obvious choice, but F also has a nice sound to it.
A final note on chords is that they can be produced by a combination of single note instruments working together rather than just the chord-capable instruments.
Melody - this aspect of music is responsible for conveying meaning and emotion. It tends to be the most memorable and enjoyable part of music, the part you sing or hum when the music is turned off. There are dual melodies such as with counterpoint. In the case of harmonizing a melody, I am simply considering that just another melody, one that tends to take a slave role to the master melody in many cases. If you have a consistent higher third harmony, the harmony played alone will sound odd compared to the melody.
Often times in music that is well done there is an additional track or sometimes an additional instrument that comes in to fill in or a background instrument takes the lead for pauses in the main melody. The singer stops singing for 1 or more seconds, so then maybe a guitar takes the lead to fill in.
Finally, I would say that roles are not static and can be entered and exited at any time, but for most of the time, instruments fulfill the same role throughout the song.
EDIT: So I was thinking about this post, and I think there is a problem with the chords section. While in several types of music chords are being used, I think there are types of music where middle notes generally, but sometimes high ones, are in a state of flux, perhaps representing chords without playing them as a whole or maybe not even as a arpeggio riff. So I guess it brings up the question of how melodic can a "chords" section get while still being considered a chord background.